Symmetry, Integrability and Geometry: Methods and Applications SIGMA 9 (2013), 009, 31 pages
Binary Darboux Transformations in Bidifferential
Calculus and Integrable Reductions
of Vacuum Einstein Equations
Aristophanes DIMAKIS † and Folkert MU¨LLER-HOISSEN ‡
† Department of Financial and Management Engineering, University of the Aegean,
82100 Chios, Greece
E-mail: dimakis@aegean.gr
‡ Max-Planck-Institute for Dynamics and Self-Organization, 37077 Go¨ttingen, Germany
E-mail: folkert.mueller-hoissen@ds.mpg.de
Received November 12, 2012, in final form January 29, 2013; Published online February 02, 2013
http://dx.doi.org/10.3842/SIGMA.2013.009
Abstract. We present a general solution-generating result within the bidifferential calcu-
lus approach to integrable partial differential and difference equations, based on a binary
Darboux-type transformation. This is then applied to the non-autonomous chiral model,
a certain reduction of which is known to appear in the case of the D-dimensional vacuum
Einstein equations with D − 2 commuting Killing vector fields. A large class of exact so-
lutions is obtained, and the aforementioned reduction is implemented. This results in an
alternative to the well-known Belinski–Zakharov formalism. We recover relevant examples
of space-times in dimensions four (Kerr-NUT, Tomimatsu–Sato) and five (single and double
Myers–Perry black holes, black saturn, bicycling black rings).
Key words: bidifferential calculus; binary Darboux transformation; chiral model; Einstein
equations; black ring
2010 Mathematics Subject Classification: 37K10; 16E45
1 Introduction
The bidifferential calculus formalism aims to understand integrability features and solution-
generating methods for (at least a large class of) integrable partial differential or difference
equations (PDDEs) resolved from the particularities of examples, i.e., on an as far as possible
universal level [16, 17, 19]. The most basic ingredient is a graded associative algebra, supplied
with two anti-commuting graded derivations of degree one. It can and should be regarded
as a generalization (in the spirit of noncommutative geometry) of the algebra of differential
forms on a manifold, but supplied with two analogs of the exterior derivative. Once a PDDE
is translated to this framework, it is simple to elaborate its integrability conditions. In fact, it
could not be simpler.
In this framework, Darboux transformations (see [35,54,62] and the references therein) have
first been addressed in [19]. In the latter work, we had also obtained a very simple solution gene-
rating result that evolved into improved versions in recent applications [15, 18, 20]. Its relation
with Darboux transformations has been further clarified in [15] (see Appendix A therein), an
essential step toward the much more general result that we present in this work. The resulting
class of solutions is expressed in a universal way, in the sense that the corresponding formula
holds simultaneously for all integrable PDDEs possessing a bidifferential calculus formulation.
Choosing a bidifferential calculus associated with a specific PDDE, we can generate infinite
families of (soliton-like) solutions.
2 A. Dimakis and F. Mu¨ller-Hoissen
In Section 2 we recall some basics of bidifferential calculus and set up the stage in a concise
way to formulate our general result about Darboux transformations. This hardly requires pre-
vious knowledge and can be taken as an independent and self-contained step into the world of
integrable PDDEs. That this is indeed a very powerful tool, is demonstrated in the rest of this
work, where we concentrate on one of the more tricky examples of integrable PDEs.
In Section 3 we elaborate in detail the example of the non-autonomous chiral model equation
(
ρgzg
−1)
z + 
(
ρgρg
−1)
ρ = 0 (1.1)
for an m ×m matrix g, where ρ > 0 and z are independent real variables and  = ±1.1 With
 = 1, this governs the case of the stationary, axially symmetric vacuum Einstein (m = 2) and
Einstein–Maxwell (m = 3) equations in four dimensions, where we have two commuting Killing
vector fields, one spacelike and the other one asymptotically timelike (see, e.g., [2,3,48,68]). In
an analogous way, (1.1) with m > 3 appears in the dimensional reduction of (the bosonic part
of) higher-dimensional supergravity theories to two dimensions (see, e.g., [5, 25, 30, 36, 45, 73]).
With  = −1, the above equation appears in the case of vacuum solutions of Einstein’s equations
in four dimensions with two commuting spacelike Killing vector fields [2,4,34,77], describing in
particular cylindrical gravitational waves.
In Section 3, we also present a reduction condition that, imposed on the obtained family of
solutions, achieves that g is symmetric. Any such real and symmetric g determines a solution
of the vacuum Einstein equations in m + 2 dimensions, as recalled in Section 4. The resulting
recipe to construct solutions of the vacuum Einstein equations is close to the familiar Belinski–
Zakharov method [2–4], which has been applied in numerous publications. In a sense, what we
obtained is a kind of matrix version of the latter.
A well-known reformulation of the equations obtained from the integrable reduction of the
four-dimensional vacuum Einstein (and also the Einstein–Maxwell) equations by introduction
of the so-called twist potential (a crucial step toward the Ernst equation), connects space-time
metrics in a different way with the non-autonomous chiral model (see, e.g., [58]). Solutions
of the latter are then required to have a constant determinant. This allows a constant seed
and thus an application of the restricted solution-generating result in [15] (also see Remark 4.3
below). It does not work in the more direct approach we take in the present work, but here we
resolve the restriction in [15].
In Section 4 we elaborate several examples in four and five space-time dimensions, and show
that the resulting metrics include important solutions of Einstein’s equations, in particular some
of the more recently found black objects in five dimensions, which have no counterpart in four
dimensions (see, e.g., [25, 42,45,73]). Here we used Mathematica2 in a substantial way.
Section 5 contains some concluding remarks.
2 Binary Darboux transformations in bidifferential calculus
A graded associative algebra is an associative algebra Ω over C with a direct sum decomposi-
tion Ω =
⊕
r≥0 Ω
r into a subalgebra A := Ω0 and A-bimodules Ωr, such that ΩrΩs ⊆ Ωr+s.
A bidifferential calculus (or bidifferential graded algebra) is a unital graded associative algebra Ω,
equipped with two (C-linear) graded derivations3 d, d¯ : Ω→ Ω of degree one (hence dΩr ⊆ Ωr+1,
d¯Ωr ⊆ Ωr+1), with the properties
d2 = d¯2 = dd¯ + d¯d = 0. (2.1)
1A subscript indicates a partial derivative with respect to the corresponding variable. Formally, ρ 7→ iρ relates
the two values of  in (1.1).
2Mathematica Edition: Version 8, Wolfram Research, Inc., Champaign, Illinois, 2010.
3Hence d and d¯ both satisfy the graded Leibniz rule d(χχ′) = (dχ)χ′ + (−1)rχdχ′, for all χ ∈ Ωr and χ′ ∈ Ω.
Binary Darboux Transformations and Vacuum Einstein Equations 3
In the bidifferential calculus approach to integrable PDDEs we are looking for a choice
(Ω, d, d¯) and some φ ∈ A, or some invertible g ∈ A, such that either
d¯dφ = dφdφ (2.2)
or
d
[
(d¯g)g−1
]
= 0 (2.3)
is equivalent to a certain PDDE. The two equations are related by the Miura equation
(d¯g)g−1 = dφ, (2.4)
which establishes a kind of Miura transformation4 between the two equations (2.2) and (2.3).
This equation has both, (2.2) and (2.3), as integrability conditions. As a consequence, if we find
a solution pair (φ, g) of (2.4), then φ solves (2.2) and g solves (2.3).
In the following theorem, we formulate a solution-generating result which amounts to a trans-
formation that takes a given solution (φ0, g0) of the Miura equation (2.4) to a new solution
of (2.4),
(φ0, g0) 7→ (φ, g).
This induces corresponding transformations of solutions of (2.2), respectively (2.3).
Let nowA be the algebra of all finite-dimensional5 matrices, with entries in a unital algebra B.
The product of two matrices is defined to be zero if the sizes of the two matrices do not match.
In the following we assume that there is a graded algebra Ω with Ω0 = A, and a bidifferential
calculus (Ω,d, d¯), and such that d and d¯ preserve the size of matrices. I = Im and I = In
denote the m×m, respectively n×n, identity matrix, and we assume that they are annihilated
by d and d¯. Furthermore, Mat(m,n,B) denotes the set of m× n matrices over B.
Theorem 2.1. Let φ0, g0 ∈ Mat(m,m,B) solve the Miura equation (2.4). Let P ,Q ∈
Mat(n, n,B) be invertible solutions of
d¯P = (dP )P , d¯Q = QdQ, (2.5)
which are independent in the sense that QY = Y P implies Y = 0. Let U ∈ Mat(m,n,B) and
V ∈ Mat(n,m,B) be solutions of the linear equations6
d¯U = (dU)P + (dφ0)U , d¯V = QdV − V dφ0. (2.6)
Furthermore, let X ∈ Mat(n, n,B) be an invertible solution of the Sylvester-type equation
XP −QX = V U . (2.7)
Then
φ = φ0 + UX−1V , g =
(
I + U(QX)−1V
)
g0 (2.8)
solve the Miura equation (2.4), and thus also (2.2), respectively (2.3).
4The original Miura transformation maps solutions of the modified Korteweg–de Vries equation to solutions
of the Korteweg–de Vries equation.
5An extension to a suitable class of ∞-dimensional matrices, respectively operators, is certainly possible.
6If we are primarily interested in solving (2.3), it is more convenient to replace dφ0 by (d¯g0)g−10 in these
equations, by use of the Miura equation for (φ0, g0).
4 A. Dimakis and F. Mu¨ller-Hoissen
Proof. Acting on (2.7) with d¯, using the rules of bidifferential calculus, (2.5), (2.6), and (2.7)
again, we find
Q[d¯X − (dX)P + (dQ)X + (dV )U ] = [d¯X − (dX)P + (dQ)X + (dV )U ]P .
Since P and Q are assumed to be independent, this implies7
d¯X − (dX)P + (dQ)X + (dV )U = 0. (2.9)
Using also (2.5) and (2.7), we easily deduce that
d¯(QX)−1 = −X−1
[
(dX)X−1(XP )− (dV )U
]
(QX)−1
= (dX−1)
[
I + V U(QX)−1
]
+ X−1(dV )U(QX)−1.
This is then used in the elaboration of
d¯g = d¯
[
(I + U(QX)−1V )g0
]
via the graded derivation (Leibniz) rule for d¯. We eliminate all further terms involving a d¯ with
the help of (2.6), and apply (2.7) to eliminate a P in favor of a Q. Finally we obtain d¯g = (dφ)g
with φ given by (2.8). Our assumptions ensure that the inverse of g exists. It is given by
g−1 = g−10
(
I −U(XP )−1V
)
.
The integrability conditions of (2.6) and (2.9) are satisfied as a consequence of (2.5) and d¯dφ0 =
dφ0dφ0 (which follows from (d¯g0)g
−1
0 = dφ0). 
For a reader acquainted with corresponding results about binary Darboux transformations in
the literature on integrable systems, the similarity will be evident (see, e.g., [12, 35, 54, 59, 62]).
It is close to results in [63–65]8. A new feature, however, is the dependence of the linear
equations (2.6) on solutions P and Q of the nonlinear equations (2.5). This generalization is
crucial for our application to the Einstein equations. As already mentioned in the introduction,
Theorem 2.1 considerably generalizes previous, more restricted results, see in particular [15].
Remark 2.1. As a consequence of (2.5), P and −Q solve the n× n version of (2.2). Since we
assume that they are invertible, P and Q−1 solve the n× n version of (2.3). If U is a solution
of the first of (2.6), then also UP , by use of the first of (2.5). If V is a solution of the second
of (2.6), then also QV , by use of the second of (2.5).
Remark 2.2. By direct computations one verifies the following “scaling property”. Let U
and V solve (2.6) and let
gˆ0 = wg0, Uˆ = UW 1, Vˆ = W 2V ,
with a solution (ϕ,w) of the m = 1 version of (2.4), and invertible W i ∈ Mat(n, n,B), subject
to
[W 1,P ] = 0, [W 2,Q] = 0.
We further assume that ϕ and w are in the center of Ω.9 If W 1 and W 2 satisfy
d¯W 1 = (dW 1)P + (d¯w)w−1W 1, d¯W 2 = QdW 2 −W 2(d¯w)w−1,
7It is a key feature of Theorem 2.1 that this equation is automatically solved if P and Q are independent.
Relaxing the latter assumption, the theorem remains valid if this equation is added to the assumptions.
8The precise relation has still to be clarified.
9This holds, e.g., in the case of the non-autonomous chiral model treated in Section 3. But this assumption
will be a (perhaps too) severe restriction if B involves differential or difference operators.
Binary Darboux Transformations and Vacuum Einstein Equations 5
then Uˆ and Vˆ solve (2.6), with φ0 replaced by φˆ0 = φ0 + ϕI. Furthermore, Xˆ = W 2XW 1
solves (2.7) with U , V replaced by Uˆ , Vˆ . For the new solutions of (2.4) (and thus (2.2),
respectively (2.3)), determined by the theorem, we find
φˆ = φ+ ϕI, gˆ = wg,
where φ and g are given by (2.8). Hence, under the stated conditions, multiplication of g0 by
a “scalar” solution w of (2.3) simplify amounts to multiplication of g by w. In the special case
where w is the identity element and g thus remains unchanged, the above equations for W 1
and W 2 still allow non-trivial transformations of U and V . These results will be used in
Section 4.
Remark 2.3. Let (φi, gi), i = 1, 2, be two solutions obtained via Theorem 2.1 from the same
seed solution (φ0, g0). Let (P i,Qi,U i,V i) be corresponding solutions of (2.5) and (2.6). Then
P =
(
P 1 0
0 P 2
)
, Q =
(
Q1 0
0 Q2
)
, U = (U1,U2), V =
(
V 1
V 2
)
also solve (2.5) and (2.6). If P and Q are independent and if (2.7) has an invertible solution X,
then (2.8) determines a new solution of (2.4), and thus new solutions of (2.2) and (2.3). This
expresses a nonlinear superposition principle. By iteration, one can build superpositions of an
arbitrary number of “elementary” solutions, hence (analogs of) “multi-solitons”.
Remark 2.4. We note that the first of equations (2.6) is a special case (and n × n matrix
version) of the general linear equation
d¯ψ − Aψ = dψP, where A = dφ. (2.10)
Here P is a solution of the nonlinear equation
d¯P = (dP )P.
An n× n matrix version of it appears in (2.5). The integrability condition of (2.10) is (2.2). If
we set A = (d¯g)g−1, the integrability condition is (2.3) instead. A similar statement holds for
the second of equations (2.6), the “adjoint linear system”. The Darboux transformation (2.8) is
“binary” since it involves solutions of the linear system as well as of the adjoint.
Remark 2.5. In the special case where Ω is the algebra of matrix-valued differential forms
on a manifold, and d the exterior derivative, any tensor field N of type (1, 1) with vanishing
Nijenhuis torsion determines a map d¯N satisfying the above conditions [7, 31]. Moreover, ac-
cording to Fro¨licher–Nijenhuis theory [31], any d¯ such that (2.1) holds has to be of this form.
Finite-dimensional integrable systems have been considered in this framework in [13]. The ge-
neralization to Lie algebroid structures is nicely described in [6]. See also [1, 51, 52] for related
aspects. We should stress, however, that in our central examples we depart from differential
geometry and consider a differential calculus in a weaker sense (e.g., of noncommutative geo-
metry).
The setting of the above theorem allows in principle structures far away from classical calculus
and in particular differential geometry, then dealing with equations beyond differential and
difference equations. A somewhat more restricted framework is given by setting
Ω = A⊗
∧(
CN
)
,
where
∧
(CN ) denotes the exterior (Grassmann) algebra of the vector space CN . In this case
it is sufficient to define suitable operators d and d¯ on A, since they extend to Ω in an evident
way. Many integrable PDDEs have been treated in this framework and in the next section we
turn to an important example.
6 A. Dimakis and F. Mu¨ller-Hoissen
3 Solutions of the non-autonomous chiral model
The non-autonomous chiral model is well-known to be a reduction of the (anti-) self-dual Yang–
Mills equations, for which a very simple bidifferential calculus exists [19] (that may actually
be considered as a prototype). From the latter one can then derive a bidifferential calculus
for (1.1). It is determined by
df = −fzξ1 + e
θ(fρ − ρ
−1fθ
)
ξ2, d¯f = e
−θ(fρ + ρ
−1fθ
)
ξ1 + fzξ2
for f ∈ C∞(R3) (cf. [15]). ξ1, ξ2 is a basis of
∧1(C2). Choosing B = C∞(R3), d and d¯ extend
to A⊗
∧
(C2) via d(f1ξ1 + f2ξ2) = (df1) ∧ ξ1 + (df2) ∧ ξ2 and d(fξ1 ∧ ξ2) = (df) ∧ ξ1 ∧ ξ2 = 0,
and correspondingly for d¯. For an m×m matrix-valued function g, (2.3) now takes the form
(
ρgzg
−1)
z + 
(
ρgρg
−1)
ρ −
[(
gρ + ρ
−1gθ
)
g−1
]
θ +
(
gθg
−1)
ρ = 0,
which reduces to (1.1) if g does not depend on θ. The coordinate θ is needed to have the prop-
erties of a bidifferential calculus, but addressing (1.1) we are primarily interested in equations
for objects that do not depend on it.
Using this bidifferential calculus, the Miura equation (2.4) decomposes into
(
gρ + ρ
−1gθ
)
g−1 = −eθφz, gzg
−1 = eθ
(
φρ − ρ
−1φθ
)
.
If g is θ-independent, this requires
φ = e−θφ˜, (3.1)
with θ-independent φ˜, and then reduces to
gρg
−1 = −φ˜z, gzg
−1 = φ˜ρ + ρ
−1φ˜. (3.2)
In the following, we elaborate Theorem 2.1 using the above bidifferential calculus. Section 3.1
provides the complete solution of the two nonlinear equations (2.5) under the condition that P
(respectively Q) has geometrically simple spectrum (i.e., for each eigenvalue there is a unique
corresponding Jordan block in the Jordan normal form), also see [15]. In this case the two
equations actually coincide. Then it only remains to solve linear equations, see Section 3.2 and
Section 3.3 below. In Section 3.4 we present a condition to be imposed on the data in order
to achieve that the solution g is symmetric (or Hermitian). This reduction is crucial in the
context of Einstein’s equations, see Section 4. Most of the following is, however, independent
of additional assumptions and provides a general procedure to construct solutions of the non-
autonomous chiral model. The corresponding equations cannot be solved explicitly without some
restrictions, in particular on the form of the seed solution. We content ourselves with providing
illustrative and important examples relevant in the context of gravity in Section 4.
3.1 The equations for P and Q
In terms of
P˜ := eθP ,
the first of the two equations (2.5) decomposes into the following pair of equations,
P˜ ρ + ρ−1
(
P˜ θ − P˜
)
= −P˜ zP˜ , P˜ z =
[
P˜ ρ − ρ−1
(
P˜ θ − P˜
)]
P˜ , (3.3)
Binary Darboux Transformations and Vacuum Einstein Equations 7
which are autonomous in the variable θ. Assuming that P˜ and I + P˜
2
are invertible, and that
P˜ is θ-independent, (3.3) implies
P˜
2
− 2ρ−1(zI −A)P˜ − I = 0, (3.4)
with an arbitrary constant n × n matrix A (also see [15]). This is a matrix version of the
pole trajectories in the Belinski–Zakharov approach [2,4]. A well-known symmetry of the latter
extends to the matrix case: (3.4) is invariant under P˜ 7→ −P˜
−1
. For the following result, see
Lemma 4.1 in [15].
Lemma 3.1. Any θ-independent, invertible solution of (3.4), which commutes with its deriva-
tives with respect to ρ and z, solves (3.3).
A can be taken in Jordan normal form A = block-diag(An1 , . . . ,Ans), without restriction of
generality, and a solution of (3.4) is then given by
P˜ = block-diag(P˜ n1 , . . . , P˜ ns),
where the block P˜ ni is a solution of (3.4) with A replaced by the Jordan block Ani , see the next
examples. Under the assumption that P˜ has geometrically simple spectrum, this is the most
general solution of d¯P = (dP )P with θ-independent P˜ = eθP [15].
Example 3.1. If A is diagonal, i.e., A = diag(a1, . . . , an) with constants ai, corresponding
solutions of (3.4) are given by P˜ = diag(p˜1, . . . , p˜n), where p˜i is any of
pi = ρ
−1(z − ai + Ri), p¯i = ρ−1(z − ai −Ri), (3.5)
with Ri =
√
(z − ai)2 + ρ2. Note that p¯i = −/pi.
Remark 3.1. For better comparison with the relevant literature, we will sometimes write
pi =
ρ
µi
, p¯i =
ρ
µ¯i
,
where
µi =
√
(z − ai)2 + ρ2 − (z − ai), µ¯i = −
√
(z − ai)2 + ρ2 − (z − ai). (3.6)
µi and µ¯i are often referred to as (indicating the presence of) a soliton and an anti-soliton,
respectively. Note that µ¯i = −ρ2/µi. If  = 1, then µi is non-negative and only vanishes on
a subset of {ρ = 0}.
Example 3.2. For an r × r Jordan block
Ar = aIr + N r, N r =









0 1 0 · · · 0
0 0 1
. . .
...
...
. . . . . .
...
...
. . . 1
0 · · · · · · · · · 0









,
(3.4) has the solutions (see [15])
P˜ r = ρ−1
(
zIr −Ar +
r−1∑
k=0
(
1/2
k
)
(±R)1−2k
[
2(a− z)N r + N2r
]k
)
,
8 A. Dimakis and F. Mu¨ller-Hoissen
where R =
√
(z − a)2 + ρ2. This is an upper-triangular Toeplitz matrix and thus commutes
with its derivatives. In particular, we have P˜ 1 = p˜ = ρ−1[z − a±R] and
P˜ 2 = p˜
(
1 ∓R−1
0 1
)
, P˜ 3 = p˜


1 ∓R−1 ± 2ρR
−3p˜−1
0 1 ∓R−1
0 0 1

 .
For the solutions P˜ obtained in this way, and thus P , we have
d¯P = (dP )P = P (dP ),
so that they also provide us with solutions Q = e−θQ˜, with θ-independent Q˜, of the second
of (2.5), also see [15].
Recall that on the way to the above results we assumed that P˜ and I + P˜
2
are invertible,
and then also Q˜ and I + Q˜
2
. These assumption will also be made throughout in the following.
Remark 3.2. The source matrix A for P˜ and the corresponding source matrix A′ for Q˜ can
be assumed to be simultaneously in Jordan normal form, without restriction of generality. Since
this is achieved by similarity transformations with constant transformation matrices, the latter
can be absorbed by redefinitions that restore all the equations obtained from Theorem 2.1. But
in general this will no longer be so if we impose a reduction condition that relates P˜ and Q˜, as
in Section 3.4 below.
3.2 The equations for U and V
Setting φ0 = e−θφ˜0 with a θ-independent φ˜0 (cf. (3.1)), (2.6) is autonomous in the variable θ.
Assuming that U , V are θ-independent, and using the fact that φ˜0 has to solve the Miura
equations (3.2) together with some g0, we obtain the following systems of linear differential
equations for U and V ,
Uρ =
(
g0,ρg
−1
0 U − g0,zg
−1
0 UP˜
)(
I + P˜
2)−1
,
U z =
(
g0,zg
−1
0 U + g0,ρg
−1
0 UP˜
)(
I + P˜
2)−1
, (3.7)
and
V ρ =
(
I + Q˜
2)−1(Q˜V g0,zg
−1
0 − V g0,ρg
−1
0
)
,
V z = −
(
I + Q˜
2)−1(V g0,zg
−1
0 + Q˜V g0,ρg
−1
0
)
. (3.8)
These equations have to be solved for the given seed solution g0 of (2.3). For diagonal g0, this
is done in the next example for the V -equations. Similar results are easily obtained for the
U -equations.
Example 3.3. Let
Q˜ = diag(p˜1, . . . , p˜n), g0 = diag(w1, . . . , wm),
where p˜i is either pi or p¯i in (3.5), and wα is a non-vanishing solution of the scalar (i.e., m = 1)
version
[ρ(lnw)ρ]ρ = −[ρ(lnw)z]z (3.9)
Binary Darboux Transformations and Vacuum Einstein Equations 9
of the non-autonomous chiral model (1.1). Writing V = (Viα), where i = 1, . . . , n and α =
1, . . . ,m, (3.8) reads
(lnViα)ρ =
1
1 + p˜2i
(p˜i(lnwα)z − (lnwα)ρ) ,
(lnViα)z = −
1
1 + p˜2i
((lnwα)z + p˜i(lnwα)ρ) .
Let us list some simple solutions. If wα is constant, then also Viα. If p˜i = pi, then
Viα = kiα
{
(ρpi)−1/2 if wα = ρ,
(ρpi)1/2
(
1 + p−1i p
−1
α
)
if wα = pα,
where pα shall be given by the same expression as some pi, but with in general different constant,
say a′α. kiα is an arbitrary constant. If p˜i = p¯i, then
Viα = kiα
{
(pi/ρ)1/2 if wα = ρ,
(pi/ρ)1/2(1 + pipα)−1 if wα = pα.
More complicated solutions are now obtained by noting the following.
• If Viα is a solution for wα, then V
−1
iα is a solution for w
−1
α .
• If wα is the product of two solutions of (3.9), then Viα is the product of the respective
solutions for the factors.
3.3 The Sylvester equation and the solution formula
Recalling that U , V and Q˜ are θ-independent, the formula for (θ-independent) g in (2.8) requires
X = eθX˜ with θ-independent X˜. (2.7) becomes the θ-independent Sylvester equation
X˜P˜ − Q˜X˜ = V U . (3.10)
If spec(P˜ ) ∩ spec(Q˜) = ∅, then (3.10) has a unique solution, for any choice of the matrices
on the right hand side. The two matrices P and Q are then independent, hence Theorem 2.1
implies that10
g =
(
I + U
(
Q˜X˜
)−1V
)
g0 (3.11)
solves the non-autonomous chiral model equation (1.1). Obviously, scaling U or V with an
arbitrary non-zero constant leaves g invariant. We recall from [15] (see Remark 4.4 therein) that
det g =
det P˜
det Q˜
det g0. (3.12)
Example 3.4. Let P˜ be diagonal, as in Example 3.1, and also Q˜, with eigenvalues q˜i, given by
an expression of the same form as p˜i. If they have no eigenvalue in common, i.e., {p˜i}∩{q˜i} = ∅,
then the unique solution of (3.10) is given by the Cauchy-like matrix
X˜ij =
(V U)ij
p˜j − q˜i
.
10We have to choose Q˜ invertible and make sure that the solution X˜ of (3.10) is invertible. See [14, 38] for
conditions that guarantee the latter.
10 A. Dimakis and F. Mu¨ller-Hoissen
A vast literature exists on solutions of the Sylvester equation (3.10), more generally with
non-diagonal matrices P˜ and Q˜ (and not necessarily satisfying the spectrum condition that
guarantees a unique solution).
Proposition 3.1 ( [14,37,44]). Let spec(P˜ ) ∩ spec(Q˜) = ∅ and
P(λ) =
n∑
k=0
Pkλk
be the characteristic polynomial of P˜ . Then the unique solution of the Sylvester equation (3.10)
is given by
X˜ = −P(Q˜)−1
n∑
k=1
Pk
k−1∑
i=0
Q˜
k−1−i
V UP˜
i
. (3.13)
Remark 3.3. (2.9) takes the form
X˜ρ + ρ−1X˜ + X˜zP˜ − Q˜zX˜ − V zU = 0,
X˜z −
(
X˜ρ − ρ−1X˜
)
P˜ +
(
Q˜ρ + ρ
−1Q˜
)
X˜ + V ρU = 0. (3.14)
If we drop the spectrum condition for P˜ and Q˜, these equations also have to be solved. Otherwise
they are a consequence of our assumptions (see the proof of Theorem 2.1). (3.14) will only be
used in the proof of Proposition 4.1 in Appendix B.
Remark 3.4. Using the above results, we also obtain solutions φ of (2.2), given by the expression
in (2.8), which in the case under consideration and via (3.1) takes the form
φ˜zz +
(
φ˜ρ + ρ
−1φ˜
)
ρ =
[
φ˜ρ + ρ
−1φ˜, φ˜z
]
(which corrects a typo in (4.3) of [15]).
3.4 A reduction condition
It is of particular interest (see Section 4) to find a convenient condition which guarantees that
the solution matrix g given by (3.11) is symmetric, i.e., gᵀ = g, where ᵀ means matrix transpose.
The following result is easily verified by a direct computation.
Lemma 3.2. Let P˜
ᵀ
= −Q˜
−1
and gᵀ0 = g0. If V solves (3.8), then U = (V g0)
ᵀ solves (3.7).
Proposition 3.2. Let P˜
ᵀ
= −Q˜
−1
, spec(P˜ )∩ spec(Q˜) = ∅, gᵀ0 = g0, and U = (V g0)
ᵀ. Then
g given by (3.11) is symmetric.
Proof. Using gᵀ0 = g0 and U = (V g0)
ᵀ, the Sylvester equation (3.10) and its transpose lead to
Q˜
[
Q˜X˜ − (Q˜X˜)ᵀ
]
= Q˜
[
X˜P˜ − P˜
ᵀ
X˜
ᵀ]
=
[
Q˜X˜ + X˜
ᵀ
P˜
−1]
P˜ =
[
Q˜X˜ − (Q˜X˜)ᵀ
]
P˜ .
In the last two steps we used P˜
ᵀ
= −Q˜
−1
. Now the spectrum condition implies Q˜X˜ = (Q˜X˜)ᵀ.
Together with gᵀ0 = g0 and U = (V g0)
ᵀ, this shows that g given by (3.11) is symmetric. 
Remark 3.5. Lemma 3.2 and Proposition 3.2 also hold with transposition replaced by any
involutory anti-automorphism of the matrix algebra, hence in particular for Hermitian conju-
gation. The Hermitian reduction of the non-autonomous chiral model appears in particular in
the context of the (electro-vacuum) Einstein–Maxwell equations in four dimensions with two
commuting Killing vector fields (also see [15]).
Binary Darboux Transformations and Vacuum Einstein Equations 11
In terms of the matrix
Γ := −QX˜ = 
(
P˜
ᵀ)−1X˜, (3.15)
which is symmetric under the assumptions of Proposition 3.2 (see the proof of Proposition 3.2),
the Sylvester equation (3.10) takes the form of a Stein equation,
Γ + P˜
ᵀ
ΓP˜ = V g0V ᵀ. (3.16)
Implementing the assumptions of Proposition 3.2 in the solution formula (3.11), we have
g =
(
I − g0V ᵀΓ−1V
)
g0. (3.17)
If P˜ is diagonal, i.e., P˜ = diag(p˜1, . . . , p˜n), and if p˜ip˜j 6= − for all i, j, then the solution of (3.16)
is given (via Example 3.4) by
Γij = 
(V g0V ᵀ)ij
p˜ip˜j + 
. (3.18)
This is essentially the corresponding matrix (usually denoted by Γ) in the Belinski–Zakharov
method [2, 3].
Remark 3.6. From Remark 2.3 we deduce the following superposition result. Let (P˜ i,V i), i =
1, . . . , N , be solutions of (3.3) and (3.8) with Q˜i = −(P˜
ᵀ
i )
−1, for the same seed solution g0. Then
P˜ = block-diag(P˜ 1, . . . , P˜N ) and V = (V
ᵀ
1, . . . ,V
ᵀ
N )
ᵀ solve (3.3) and (3.8) with Q˜ = −(P˜
ᵀ
)−1.
If spec(P˜ )∩ spec(−(P˜
ᵀ
)−1) = ∅, and if (3.16) has an invertible solution Γ, then (3.17) is again
a symmetric solution of (1.1).
4 Solutions of the vacuum Einstein equations
In D dimensions, let us consider a space-time metric of the form
ds2 = gαβdx
αdxβ + f
(
dρ2 + dz2
)
,
where the (real) components gαβ, α, β = 1, . . . ,m, and the function f only depend on the
coordinates ρ and z (and thus not on x1, . . . , xm). The metric then obviously admits m = D−2
commuting Killing vector fields11. For the next result, see, e.g., [2, 5, 25,73]12. If
(1) the matrix g = (gαβ) satisfies det g = −ρ2,
(2) g solves the m×m non-autonomous chiral model equation (1.1),
(3) f is a solution of the compatible system of linear equations
(ln f)ρ = −
1
ρ
+
1
4ρ
tr
(
U2 − V2
)
, (ln f)z =
1
2ρ
tr(UV), (4.1)
where U := ρgρg−1 and V := ρgzg−1,
11It is not the most general metric admitting m = D − 2 commuting Killing vector fields. See, e.g., [2].
12There are also reductions of the Einstein vacuum equations to the non-autonomous chiral model equation
using a non-Abelian Lie algebra of Killing vector fields. See the case with a null (i.e., lightlike, or isotropic) Killing
vector field in four dimensions treated in [67].
12 A. Dimakis and F. Mu¨ller-Hoissen
then the metric is Ricci-flat, hence a solution of the vacuum Einstein equations (with vanishing
cosmological constant).
Since g has to be real and symmetric, corresponding conditions have to be imposed on the
matrix data of the class of solutions obtained in Section 3, so that these solutions determine Ricci-
flat metrics (also see [15]). Such conditions have been found in Section 3.4, and, accordingly, in
all examples of this section we shall set
Q˜ = −
(
P˜
−1)ᵀ
, U = (V g0)ᵀ. (4.2)
Typically we will regard these equations as defining Q˜ and U in terms of P˜ , V and g0. In most
of the examples below, P˜ is diagonal. The (symmetric) solution of the non-autonomous chiral
model is then given by (3.17) with Γ in (3.18). If P˜ is not diagonal, we have to proceed via
the solution (3.13) of the Sylvester equation (still assuming that the spectrum condition holds)
and (3.15).
According to the following remark, the determinant condition (1) can always be achieved
if m is odd. There is a slight restriction if m is even.
Remark 4.1. By taking the trace of (1.1), one finds that det g and any power of it is a solution
of (1.1) in the scalar case (also see [2,4]). We further note that gˆ = wg, with any non-vanishing
solution w of the scalar equation, is again a solution of (1.1). For a solution g given by (3.17),
using (3.12) and (4.2) we find that
w =
(
ρ2
(det P˜ )2(−1)n+1det g0
)1/m
achieves that det gˆ = −ρ2. Since w has to be real, for even m this requires (−1)n(−det g0) > 0.
In the next subsection, we address (4.1). Then we sketch a useful procedure to construct
non-diagonal metrics from diagonal ones in such a way that the diagonal metric is recovered by
setting some parameters to zero. A collection of relevant examples in four and five space-time
dimensions follows. The method is indeed of most interest for D = 4 and D = 5. The higher
the number of dimensions, the more restrictive is the assumption of D − 2 commuting Killing
vector fields for the set of solutions of the vacuum Einstein equations.
4.1 Solutions of the equations for the metric function f
Example 4.1. For a diagonal solution g0 of (1.1), so that (g0)αα solves (3.9), (4.1) leads to
f0 = κρ
−1
m∏
α=1
fα,
where κ is an arbitrary constant and fα has to be a solution of
(ln f)ρ =
ρ
4
(
(lnw)2ρ − (lnw)
2
z
)
, (ln f)z =
ρ
2
(lnw)ρ(lnw)z, (4.3)
with w replaced by (g0)αα. If w is a constant, then also f. Let us write f[w] for the solution of
the above equations for a given solution w of (3.9), and let ∝ denote equality up to a non-zero
constant factor. In particular, we have
f[µ˜i] ∝
µ˜i
√
µ˜2i + ρ
2
, where µ˜i = ±
√
(z − ai)2 + ρ2 − (z − ai)
Binary Darboux Transformations and Vacuum Einstein Equations 13
(cf. (3.6)). More generally, we find
f
[
ρk
µ˜1 · · · µ˜r
µ˜′1 · · · µ˜
′
s
]
∝ ρk
2/4
(
µ˜1 · · · µ˜r
µ˜′1 · · · µ˜
′
s
)k/2
(
r∏
i=1
f[µ˜i]
)

s∏
j=1
f[µ˜′j ]


(
∏
i<k
F[µ˜i, µ˜k]
)
×


∏
j<l
F[µ˜′j , µ˜
′
l]


r∏
i
s∏
j
F[µ˜i, µ˜′j ]
−1,
where µ˜′i is µ˜i with the constant ai replaced by some constant a
′
i, and we introduced the abbre-
viation
F[µ˜i, µ˜j ] =
µ˜iµ˜j
µ˜iµ˜j + ρ2
=
µ˜i − µ˜j
2(ai − aj)
,
where the last equality holds if ai 6= aj . The µ˜i (and also the µ˜′i) need not be distinct in the
above formula. Using these results in our formula for f0, we have the solution of (4.1) for a large
class of diagonal solutions of (1.1). See Appendix A for some details, and also [3].
For the proofs of the following propositions, see Appendix B.
Proposition 4.1. Let f0 be a solution of (4.1) for a seed g0 (solution of the Miura equation).
Let g be a corresponding solution of the non-autonomous chiral model from the family obtained
in Section 3 (i.e., given by (3.11), where the ingredients are subject to the respective equations).
Then
f = κf0ρ
−n det P˜ det Q˜det X˜
[
det
(
I + P˜
2)
det
(
I + Q˜
2)]1/2
,
with an arbitrary constant κ, solves (4.1) with the right hand sides evaluated with g.
Proposition 4.1 does neither assume that g0 and g are symmetric, nor the reduction condi-
tions (4.2). The following corollary now specializes to the case of this reduction.
Corollary 4.1. Let g0 be symmetric, Q˜
ᵀ
= −P˜
−1
, spec(P˜ ) ∩ spec(Q˜) = ∅, and U = (V g0)ᵀ.
Then the solution of (4.1) corresponding to g given by (3.17) is
f = κf0ρ
−n (det P˜ )
2
det
(
I + P˜
2) det Γ,
with an arbitrary constant κ.
A convenient formula for the determinant of the solution Γ of the Stein equation seems not
to be available in the literature.
If we have to modify a solution g in order to achieve the determinant condition (1), the
following result is of great help.
Proposition 4.2. Let f be a solution of (4.1) for a given solution g from the class obtained
in Section 3, and let f(w) be a solution of (4.1) for a scalar solution w of the non-autonomous
chiral model (1.1). Then
fˆ = (ρf(w))
m−1 f(w det g)
f(det g)
f (4.4)
solves (4.1) with gˆ = wg (which also solves (1.1) according to Remark 4.1). If det gˆ = −ρ2,
this reduces to
fˆ =
wf
(ρf(w))m
. (4.5)
14 A. Dimakis and F. Mu¨ller-Hoissen
Due to these results, the problem of explicitly computing solutions of the vacuum Einstein
equations, based on the results in Section 3 and with a diagonal seed, essentially boils down
to that of solving the linear equations (3.8) for V . For diagonal Q˜ = −P˜
−1
, solutions of the
latter have already been obtained in Example 3.3. Although we do not yet have general results
in case of a non-diagonal Q˜, corresponding examples are treated below in Examples 4.3 and 4.4.
4.2 From diagonal to non-diagonal solutions
In the following, we impose the reduction conditions (4.2), which determine Q˜ in terms of P˜ ,
and U in terms of V and g0, and restrict our considerations to the case  = 1.
We translate a procedure due to Pomeransky [60], which has been formulated and applied
in the Belinski–Zakharov (BZ) approach, to our framework. We start with a diagonal (hence
in particular static) solution of the above dimensionally reduced vacuum Einstein equations.
The corresponding solution g˜ of (1.1) is thus diagonal. According to Remark 2.2 we may still
simplify it by multiplication with a suitable scalar solution of (1.1). Now some (anti-) solitons
are removed from it in the following way.
• Removal of a soliton pi = ρµ
−1
i (i.e., a soliton at z = ai) from g˜αα. This means multipli-
cation of g˜αα by −p
−2
i . In the BZ language, the corresponding trivial BZ vector has a 1
at the αth position and otherwise zeros.
• Removal of an anti-soliton p¯j = −p
−1
j = −ρ
−1µj (i.e., an anti-soliton at z = aj) from g˜ββ .
This means multiplication of g˜ββ by p¯
−2
j = −p
2
j .
Afterwards these solitons are reintroduced, with more freedom, in the following way.
• In order to reintroduce the corresponding soliton, set P˜ kk = pi, for some k. The kth row
of the matrix V that solves (3.8) (with Q˜
ᵀ
= −P˜
−1
), or rather the vector formed by its
constant coefficients, generalizes the aforementioned trivial BZ vector. It’s αth component
has to be non-zero.
• To reintroduce the corresponding anti-soliton, set P˜ ll = p¯j , for some l. The lth row of the
matrix V generalizes the corresponding trivial BZ vector if its βth component is non-zero.
After multiplication with a suitable scalar solution of (1.1), in order to achieve the determinant
condition, this leads to a non-diagonal (and thus typically stationary) generalization of the
original diagonal metric. We recover the latter by suitably fixing the parameters introduced in
the second step.
In particular, this offers the possibility to reconstruct a known non-diagonal solution from its
diagonal specialization (provided the latter can be achieved) via the solution-generating tech-
nique. Some well-known examples of four-dimensional space-times are recovered in Section 4.3
in this way. The above procedure will be applied in the five-dimensional case in Section 4.4.
Remark 4.2. According to an observation at the end of Remark 2.2, any transformation V 7→
WV with a constant n × n matrix W that commutes with P˜
ᵀ
(i.e., [W , P˜
ᵀ
] = 0) leaves g
given by (3.17) invariant. If P˜ is diagonal, any diagonal W commutes with it. We can then use
such a transformation of V to scale in each row of V one of the non-zero constant coefficients
to a fixed value like 1. In the above step of reintroduction of solitons, without restriction of
generality we can thus fix the constants appearing in the respective positions of the rows of V .
If P˜ = P˜ r (see Example 3.2), then W is an arbitrary constant, lower-triangular Toeplitz matrix
and we can again rescale one non-zero coefficient in each row of the corresponding matrix V
to a chosen value (different from zero), without changing the corresponding solution g. This
feature generalizes to a P˜ that is block-diagonally composed of matrices P˜ r.
Binary Darboux Transformations and Vacuum Einstein Equations 15
4.3 Solutions of the vacuum Einstein equations in four dimensions
Let D = 4, hence m = 2, and  = 1. We thus consider stationary and axially symmetric
solutions of the vacuum Einstein equations. The metric components gαβ, α, β = 1, 2, should
then refer to coordinates x1 = t (time) and x2 = ϕ (angle around the symmetry z-axis), ρ is the
coordinate distance from the axis. This interpretation in general imposes additional conditions
on the metric. The simple solution
g0 =
(
−1 0
0 ρ2
)
(4.6)
of (1.1) corresponds to the four-dimensional Minkowski metric ds2 = −dt2 + ρ2dϕ2 + dρ2 + dz2
in cylindrical coordinates. The solution f0 of (4.1) with g given by (4.6) is simply a constant,
which can be set to 1 (since this can be achieved by a coordinate transformation).
In order to facilitate comparison with the relevant literature, in the following we will mainly
use the µi introduced in (3.6), instead of the pi, see Remark 3.1. Furthermore, we will frequently
use the abbreviation (also see [73])
Rij = ρ
2 + µiµj = 2(ai − aj)
µiµj
µi − µj
,
where the last equality holds if ai 6= aj .
Example 4.2 (Kerr-NUT). In order to recover the Kerr-NUT metric, we start with its static
specialization, the Schwarzschild metric. The latter corresponds to the following solution of (1.1),
g˜ =


−
µ2
µ1
0
0 ρ2
µ1
µ2

 .
Next we remove a soliton at z = a1 from g˜11 and a soliton at z = a2 from g˜22. The resulting
matrix is simplified by rescaling it with the inverse of
w = ρ−2µ1µ2.
We obtain the seed solution
g0 = w
−1g˜ diag
(
−(ρ/µ1)
−2,−(ρ/µ2)
−2) ,
which is nothing but (4.6). Next we reintroduce the two solitons (n = 2) via
P˜ = diag
(
ρ
µ1
,
ρ
µ2
)
.
The respective solution of (3.8) (with Q˜ = −P˜
−1
) is
V =
(
v11 v12µ
−1
1
v21 v22µ
−1
2
)
,
with constants viα (see Example 3.3). According to Remark 4.2, we can set v11 = v22 = 1
without restriction of generality. Let us rename the remaining parameters in V ,
c1 = v12, c2 = v21.
16 A. Dimakis and F. Mu¨ller-Hoissen
Now we obtain the matrix Γ from (3.18). Then gˆ = wg, with g given by (3.17), satisfies the
determinant condition, i.e., det gˆ = −ρ2, and has the components
gˆ11 =
1
h
((
c21 + c
2
2
)
ρ2µ1µ2(µ1 − µ2)
2 − µ1µ2
((
1 + c21c
2
2
)
R212 − 2c1c2R11R22
))
,
gˆ12 =
(µ1 − µ2)R12
h
(
c1
(
c22µ
2
2 − ρ
2)R11 + c2
(
c21ρ
2 − µ21
)
R22
)
,
gˆ22 =
1
h
(
−
(µ1 − µ2)2
µ1µ2
(
c21ρ
8 + c22µ
4
1µ
4
2
)
+
ρ2
µ1µ2
((
µ41 + c
2
1c
2
2µ
4
2
)
R212 − 2c1c2µ
2
1µ
2
2R11R22
)
)
,
where
h = (µ1 − µ2)
2(c21ρ
4 + c22µ
2
1µ
2
2
)
+
(
µ21 + c
2
1c
2
2µ
2
2
)
R212 − 2c1c2µ1µ2R11R22.
Furthermore, according to Example 4.1, we have
f(w) ∝
µ1µ2√
R11R22R12
.
Corollary 4.1 and Proposition 4.2 then yield
fˆ = −
κ
µ1µ2R11R22
h,
with a constant κ. Setting a2 = −a1 = σ,
c1 =
m + σ
a + b
, c2 =
a + b
m− σ
, κ = C
(m− σ)2
4σ2
, σ =
√
m2 − a2 + b2,
with new constants a, b, m and C, and passing over from the coordinates ρ and z to new
coordinates r and θ via
ρ =
√
(r −m)2 − σ2 sin θ, z = (r −m) cos θ,
we obtain the Kerr-NUT metric in Boyer–Lindquist coordinates, as given by (8.48) and (8.49)
in [2]. We can compose N Kerr-NUT data blockwise into larger matrices (the new P˜ and V
are then 2N × 2N , respectively 2N × 2 matrices), and obtain again a symmetric solution of
the non-autonomous chiral model equation, see Remark 3.6. The determinant condition is again
achieved by multiplication with a suitable scalar. In this way we can recover the multi-Kerr-NUT
solutions [50].
Choosing for P˜ the non-diagonal solution of (3.4) resulting from an n × n Jordan block
matrix An (see Example 3.2), we should expect that the resulting family of solutions of the
non-autonomous chiral model can be obtained alternatively via a “soliton coincidence” limit
(“pole fusion” in the Belinski–Zakharov formalism [2]) of the family of solutions obtained with
a diagonal P˜ (with distinct eigenvalues). This is confirmed by the following example.
Example 4.3. Again, let n = 2 and g0 as in (4.6). Now we choose
P˜ =
ρ
µ1
(
1 −R−1
0 1
)
,
where µ1 = R − z with R =
√
z2 + ρ2. This is P˜ 2 in Example 3.2 (with a = 0). We find the
solution
V =
(
v11 v12µ
−1
1
v21 v22µ
−1
1 − 2v12R
−1
11
)
Binary Darboux Transformations and Vacuum Einstein Equations 17
of (3.8), with constants viα. Computing the solution of the Sylvester equation via (3.13), the
symmetric matrix Γ is obtained from (3.15). We have to rescale the resulting g, given by (3.17),
to gˆ = wg with w = ρ−2µ21, in order to arrange the determinant condition. It turns out that gˆ
only depends on the parameters viα via α = v12/v11 and β = det(viα)/v211. Furthermore, we
find that gˆ also results from the corresponding (Kerr-NUT) solution in Example 4.2 in the
limit σ → 0, after setting c1 = α and c2 = α−2(α− 2βσ).
The Kerr-NUT metric consists of two solitons of multiplicity one. The Tomimatsu–Sato
metrics [72] are known to generalize it to two solitons of multiplicity δ ≥ 1 (see, e.g., [2])13. We
should then expect that, for fixed δ, this solution can be obtained alternatively with P˜ consisting
of two blocks P˜ δ as given in Example 3.2 (corresponding to the δ × δ Jordan block Aδ). For
δ = 2 this is confirmed in the next example.
Example 4.4 (Tomimatsu–Sato). In the static limit, the δ = 2 Tomimatsu–Sato metric (also
see [49]) reduces to the Zipoy–Voorhees metric. The latter corresponds to the diagonal solution
g˜ = diag
(
−µ−21 µ
2
2, ρ
2µ21µ
−2
2
)
of (1.1). Now we remove twice a soliton at z = a1 = σ from g˜11, and also twice a soliton at
z = a2 = −σ from g˜22. With a simplifying scaling we obtain the seed
g0 = w
−1g˜ diag
(
ρ−4µ41, ρ
−4µ42
)
, w = ρ−4µ21µ
2
2.
The resulting g0 is again (4.6). Next we reintroduce the two double-solitons via14
P˜ =




ρµ−11 −2ρR
−1
11 0 0
0 ρµ−11 0 0
0 0 ρµ−12 −2ρR
−1
22
0 0 0 ρµ−12



 .
The solution of (3.8) is then given by
V =




v11 v12µ
−1
1
v21 v22µ
−1
1 − 2v12R
−1
11
v31 v32µ
−1
2
v41 v42µ
−1
2 − 2v32R
−1
22



 .
Again, the matrix Γ is obtained via (3.13) and (3.15). Next we have to compute gˆ = wg, with g
given by (3.17), and
fˆ = κρ−6wf−2(w)
(det P˜ )2 det Γ
det(I + P˜
2
)
, where f(w) = ρ
3
(
µ21µ
2
2
R11R12R22
)2
.
According to Remark 4.2, without restriction of generality we can set
v11 = v21 = v32 = v42 = 1.
Choosing
v12 = v22 = v31 = v41 =
q
1 + p
, κ = σ−2
(
1 + p
4p
)4
,
13The Tomimatsu–Sato metrics are in fact limiting cases of multiple Kerr-NUT solutions [21,50].
14Noting that p1/R1 = 2ρ/R11, this consists of two blocks of the form P˜ 2 in Example 3.2.
18 A. Dimakis and F. Mu¨ller-Hoissen
in terms of prolate spheroidal coordinates x, y, given by
ρ = σ
√
(x2 − 1)(1− y2), z = σxy, (4.7)
and after a coordinate transformation t 7→ t − 4σp−1qϕ, we obtain the δ = 2 Tomimatsu–Sato
metric
ds2 = −
A
B
dt2 + 8σ
(
1− y2
)qC
pB
dtdϕ+ σ2
(
1− y2
) D
p2B
dϕ2
+
B
p4(x2 − y2)3
(
dx2
x2 − 1
+
dy2
1− y2
)
,
with
A =
(
1− y2
)4g(Z)g(−Z),
B =
(
g(x) + q2y4
)2
+ 4q2y2
(
px3 + 1− (px+ 1)y2
)2
,
C = q2(px+ 1)y4
(
−y2 + 3
)
+
(
−2q2
(
px3 + 1
)
+ (px+ 1)g(x)
)
y2 −
(
2px3 − px+ 1
)
g(x),
D = p2q4
(
x2 − 1
)
y8 + 4q2(px+ 1)
(
p3x3 + 3p2x2 − p3x+ 4px− 3p2 + 4
)
y6
− 2q2
(
3p4x6 + 4p3x5 − 3p4x4 + 8p3x3 + 37p2x2 − 12p3x+ 48px− 13p2 + 24
)
y4
+ 4q2
(
p4x8 − p4x6 + 4p3x5 − 4p3x3 + 15p2x2 + 24px− 3p2 + 12
)
y2
+ g(x)
(
p4x6+ 6p3x5− p4x4+ 16p2x4− 12p3x3+ 32px3+ 15p2x2+ 6p3x− 15p2+ 16
)
,
where p2 + q2 = 1 and
g(x) = p2x4 + 2px3 − 2px− 1, Z2 =
x2 − y2
1− y2
.
These are the expressions (32)–(35) in [49]15.
Remark 4.3. The m = 2 non-autonomous chiral model is also related to the D = 4 vacuum
Einstein equations with two commuting Killing vector fields in another way, via the Ernst
equation (see, e.g., [68]). In this case, one has to impose a different determinant condition:
det g = 1. This allows a constant seed g0, and thus constant matrices U , V , in which case the
application of Theorem 2.1 is considerably simplified. The solutions obtained in this way include
the multi-Kerr-NUT metrics (see [15] for details)16. A similar construction with m = 3 leads
to the (electrically and magnetically) charged generalizations, the multi-Demianski–Newman
metrics [15].
4.4 Solutions of the vacuum Einstein equations in five dimensions
Let m = 3 and  = 1. In this case it seems that no relevant solutions can be obtained by
choosing as the seed a diagonal solution of the non-autonomous chiral model corresponding to
five-dimensional Minkowski space-time. It then becomes a subtle problem to choose a suitable
seed solution. Here insights about the “rod structure” of the putative axis ρ = 0 are of great
help [2,9,27,36,60]. Some important solutions that have been obtained or recovered previously in
the Belinski–Zakharov approach will now be presented in our framework. We use the procedure
outlined in Section 4.2.
15We obtained a factor q4 in the expression for D, whereas there is a factor q2 in [49]. We believe that the
latter is a typo.
16In [78, 80], solutions of the five-dimensional Einstein–Maxwell equations are constructed from a pair of solu-
tions of the m = 2 non-autonomous chiral model with symmetric g and det g = 1.
Binary Darboux Transformations and Vacuum Einstein Equations 19
4.4.1 Myers–Perry black holes
The higher-dimensional generalization of a static black hole is given by the Schwarzschild–
Tangherlini solution [71]. The corresponding diagonal solution of (1.1) is
g˜ = diag
(
−
µ1
µ2
, µ2,
ρ2
µ1
)
.
The seed g0 is obtained from g˜ as follows:
• Remove a soliton at z = a1 from g˜33.
• Remove an anti-soliton at z = a2 from g˜22.
• Multiply the resulting matrix by w−1, with w = −µ1/µ2, to achieve a simpler form.
This results in
g0 = diag
(
1,
ρ2
µ1
, µ2
)
.
According to Example 4.1, the associated solution of (4.1) is (up to a constant factor)
f0 =
µ2√
R11R22
.
Next we reintroduce the soliton and the anti-soliton via
P˜ = diag
(
ρ
µ1
,−
µ2
ρ
)
.
The first row of the 2×3 matrix V should have a non-zero entry in the third component, which
means we recreate the soliton at z = a1 in the respective diagonal component of the seed. The
second row of V should have a non-zero entry in the second component, which means we recreate
the anti-soliton at z = a2. According to Example 3.3, the solution of (3.8) (with Q˜ = −P˜
−1
) is
given by
V =




v11 v12
µ1
R11
v13
R12
µ1µ2
v21 v22
R12
ρ2
v23
µ2
R22



 ,
with constants viα. The corresponding solution of the Stein equation (3.16) is obtained
from (3.18). With g given by (3.17), the new solution
gˆ = wg
satisfies the determinant condition: det gˆ = −ρ2. We find
f(w) =
R12
ρ
√
R11R22
(up to a constant factor). Without restriction of generality, we can set v13 = v22 = 1 (see
Remark 4.2). With the restrictions and renamings
v12 = v23 = 0, c1 = v11, c2 = v21,
20 A. Dimakis and F. Mu¨ller-Hoissen
use of Corollary 4.1 and Proposition 4.2 yields
fˆ = κµ2
c21c
2
2µ
3
1µ2 − (µ1 − µ2)
2
(
µ1
(
c21µ1µ2 + c
2
2ρ
2
)
+R212
)
(µ1 − µ2)2R11R12R22
,
with a constant κ. The corresponding metric is given by
ds2 = gˆ11dt
2 + gˆ22dϕ
2 + gˆ33dψ
2 + 2gˆ12dtdϕ+ 2gˆ13dtdψ + 2gˆ23dϕdψ + fˆ
(
dρ2 + dz2
)
.
Setting
a1 = −σ, a2 = σ,
with a constant σ, in terms of prolate spheroidal coordinates x, y, given by (4.7), we have
gˆ11 = −
(
4σx+
(
a21 − a
2
2
)
y − ρ20
)
ω−1,
gˆ12 = −a1ρ20(1− y)ω
−1,
gˆ13 = −a2ρ20(1 + y)ω
−1,
gˆ22 =
1− y
4
(
4σx+ a21 − a
2
2 + ρ
2
0 + 2a
2
1ρ
2
0(1− y)ω
−1) ,
gˆ23 =
1
2
a1a2ρ20
(
1− y2
)
ω−1,
gˆ33 =
1 + y
4
(
4σx− a21 + a
2
2 + ρ
2
0 + 2a
2
2ρ
2
0(1 + y)ω
−1) ,
fˆ =
ω
8
, ω = 4σx+ (a21 − a
2
2)y + ρ
2
0.
Performing a linear transformation of the coordinates t, ϕ, ψ,
(t, ϕ, ψ) 7→
(
t− a1ϕ− a2ψ,−
a1
c2
ϕ+
a2c2
4σ
ψ,
a1c1
4σ
ϕ−
a2
c1
ψ
)
,
and setting
σ =
1
4
√
(ρ20 − a
2
1 − a
2
2)
2 − 4a21a
2
2, κ =
(a21 − a
2
2)
2 − (4σ + ρ20)
2
16σρ20
,
c21 =
∣
∣
∣
∣
4σ(ρ20 − a
2
1 + a
2
2 − 4σ)
ρ20 + a
2
1 − a
2
2 + 4σ)
∣
∣
∣
∣ , c
2
2 =
∣
∣
∣
∣
4σ(ρ20 + a
2
1 − a
2
2 − 4σ)
ρ20 − a
2
1 + a
2
2 + 4σ
∣
∣
∣
∣ ,
we recover the analogue of the Kerr metric, i.e., the Myers–Perry solution [36,56,57] of the five-
dimensional vacuum Einstein equations, precisely in the form of equation (19) in [60]. Switching
on the constants v12 and v23 (which we set to zero above), leads to a more general class of
space-times.
4.4.2 Black saturn
A black saturn [22] (also see [11, 69, 79]) is a black hole surrounded by a black ring17. In the
following, we show how the black saturn solution, originally obtained in [22], can be recovered
in our approach. Let us start with the same static solution of the non-autonomous chiral model
as in [22],
g˜ = diag
(
−
µ1µ3
µ2µ4
,
ρ2µ4
µ3µ5
,
µ2µ5
µ1
)
.
This is motivated by a rod structure analysis (also see [9, 27, 36]). A suitable seed solution is
then constructed as follows.
17Black rings are similar to black holes, but with horizon topology S1 × SD−3. They only appear in D > 4
dimensions. See [8, 10,23,24,26,28,29,32,33,40,41,43,46,47,55,61,74–76].
Binary Darboux Transformations and Vacuum Einstein Equations 21
• Remove anti-solitons at z = a1 and at z = a3 from g˜11.
• Remove a soliton at z = a2 from g˜11.
• Multiply the resulting matrix by w−1 with w = ρ
2µ2
µ1µ3
to simplify its form.
This results in
g0 = w
−1g˜ diag
(
−
ρ2µ22
µ21µ
2
3
, 1, 1
)
= diag
(
1
µ4
,
µ1µ4
µ2µ5
,
µ3µ5
ρ2
)
.
The corresponding solution of (4.1) is obtained via Example 4.1,
f0 = k
2µ3µ4µ5
R12R15R24R45√
R11R22R33R14R25R35R44R55
,
with a constant k. We have n = 3 and choose P˜ diagonal with
P˜ 11 = −µ1/ρ, P˜ 22 = ρ/µ2, P˜ 33 = −µ3/ρ.
The solution of the equations for V is then given by
V =








v11
R14
µ1
v12
R12R15
R11R14
v13
ρ2µ1
R13R15
v21
µ2µ4
R24
v22
µ2µ5R12R24
µ1µ4R22R25
v23
µ1R23R25
µ22µ3µ5
v31
R34
µ3
v32
R23R35
R13R34
v33
ρ2µ3
R33R35








,
with constants viα. Without restriction of generality, we can set
v11 = v21 = v31 = 1.
The case considered in Section 2.2 of [22] should then correspond to the subclass of solutions
given by
v12 = v22 = v33 = 0.
The authors of [22] then only elaborate the special case
v32 = 0
further. In this case, however, the removal of the anti-soliton at z = a3 from the original
static metric and the subsequent reintroduction via P˜ 33 is actually redundant. This means that
the black saturn space-time can already be obtained from n = 2 data. The solution Γ of the
Stein equation (3.16) is given by (3.18). With g given by (3.17), we set gˆ = wg to satisfy the
determinant condition. Then gˆ reduces to g˜ if v13 = v23 = 0, as expected. We find
f(w) =
R12R23√
R11R22R33R13
(up to a constant factor) and, using Corollary 4.1 and Proposition 4.2,
fˆ = −κρ−6wf0f
−3
(w)
(det P˜ )2 det Γ
det
(
I + P˜
2) ,
22 A. Dimakis and F. Mu¨ller-Hoissen
which turns out to be a lengthy expression. Setting
v13 = 2c1(a1 − a5), v23 = c2[4(a2 − a3)(a2 − a4)]
−1, κ = 4
(a1 − a2)2(a2 − a3)2
(a1 − a3)2
,
with constants ci, we obtain the metric
ds2 = −
H2
H1
(
dt+
ω
H2
dψ
)2
+H1
(
k2P(dρ2 + dz2) +
G1
H1
dϕ2 +
G2
H2
dψ2
)
,
where
G1 =
ρ2µ4
µ3µ5
, G2 =
µ3µ5
µ4
, P = R15R
2
34R45,
H1 = F
−1 (M0 + c
2
1M1 + c
2
2M2 + c1c2M3 + c
2
1c
2
2M4
)
,
H2 =
µ3
µ4F
(
µ1
µ2
M0 − c
2
1
ρ2
µ1µ2
M1 − c
2
2
µ1µ2
ρ2
M2 + c1c2M3 + c
2
1c
2
2
µ2
µ1
M4
)
,
F = µ1µ5(µ1 − µ3)
2(µ2 − µ4)
2
(
5∏
i=1
Rii
)
R13R14R23R24R25R35,
with
M0 = µ2µ
2
5(µ1 − µ3)
2(µ2 − µ4)
2R212R
2
14R
2
23,
M1 = ρ
2µ21µ2µ3µ4µ5(µ1 − µ2)
2(µ1 − µ5)
2(µ2 − µ4)
2R223,
M2 = ρ
2µ21µ
−1
2 µ3µ4µ5(µ1 − µ2)
2(µ1 − µ3)
2R214R
2
25,
M3 = 2µ
2
1µ3µ4µ5(µ1 − µ3)(µ1 − µ5)(µ2 − µ4)R11R22R14R23R25,
M4 = µ
4
1µ
−1
2 µ
2
3µ
2
4(µ1 − µ5)
2R212R
2
25,
ω =
2
F
√
G1
(
c1R1
√
M0M1 − c2R2
√
M0M2 + c
2
1c2R2
√
M1M4 − c1c
2
2R1
√
M2M4
)
,
and Ri =
√
(z − ai)2 + ρ2. This is the black saturn metric18 as given in [22, 73], with some
obvious changes in notation, but some deviations in the factors µi in the expressions for M2, M3
and M4.
4.4.3 Double Myers–Perry black hole solution
In order to recover the double Myers–Perry black hole solution obtained in [39], we start with
the matrix that determines a static two-black hole solution (cf. (3.1) in [70]),
g˜ = diag
(
−
µ1µ4
µ2µ5
,
ρ2µ3
µ1µ4
,
µ2µ5
µ3
)
.
Removal of two solitons µ2, µ5 and two anti-solitons µ1, µ4 from g˜11, and simplification with
a suitable factor, leads to
g0 = w
−1g˜ diag
(
−
(
µ21µ
2
4
µ22µ
2
5
)−1
, 1, 1
)
= diag
(
−1,
ρ2µ3
µ2µ5
,
µ1µ4
µ3
)
, w =
µ2µ5
µ1µ4
.
18Additional conditions have to be imposed on the remaining parameters in order to achieve asymptotic flatness
and absence of naked and conical singularities, see [11,22].
Binary Darboux Transformations and Vacuum Einstein Equations 23
According to Example 4.1, the corresponding solution of (4.1) is given by
f0 =
µ1µ4
µ3



5∏
i=1
i 6=3
√
Rii



−1
R13R23R34R35
R14R25R33
.
We have n = 4 and reintroduce the removed solitons and anti-solitons via
P˜ = diag
(
−ρ−1µ1,−ρ
−1µ4, ρµ
−1
2 , ρµ
−1
5
)
.
The solution of (3.8) is given by
V =












v11 v12
R12R15
ρ2R13
v13
µ1R13
R11R14
v21 v22
R24R45
ρ2R34
v23
µ4R34
R14R44
v31 v32
µ2µ5R23
µ3R22R25
v33
µ3R12R24
µ1µ2µ4R23
v41 v42
µ2µ5R35
µ3R25R55
v43
µ3R15R45
µ1µ4µ5R35












.
In order to recover the double Myers–Perry black hole solution presented in [39], we reduce the
set of solutions obtained in this way by restricting V to
V =








1 v12
R12R15
ρ2R13
0
1 v22
R24R45
ρ2R34
0
1 0 0
1 0 0








.
We immediately notice that this means in particular “trivializing” the solitons at z = a2 and
z = a5. Hence, the solution obtained with this special choice of V can already be obtained from
n = 2 data. We should expect, however, a 4-soliton transformation to be necessary in order to
generate a (sufficiently general) double black hole solution, which suggests to explore the above
more general solution. This will not be done here, and we return to the special case with the
above restricted V . Again, we obtain Γ from (3.18). Let gˆ be the resulting solution (3.17),
multiplied by w to achieve the determinant condition. From Example 4.1, we obtain
f(w) = ρ
−1



5∏
i=1
i 6=3
√
Rii



−1
R12R15R24R45
R14R25
,
and then fˆ via Corollary 4.1 and Proposition 4.2. Setting
v12 =
b(a1 − a3)
2(a1 − a2)(a1 − a5)
, v22 =
c(a3 − a4)
2(a2 − a4)(a4 − a5)
,
κ =
(
4
(a1 − a2)(a1 − a5)(a2 − a4)(a4 − a5)
(a1 − a4)(a2 − a5)
)2
,
with constants b, c, this results in the metric
ds2 = −
H2
H1
(
dt+
ω
H2
dϕ
)2
+
ρ2µ3H1
µ2µ5H2
dϕ2 +
µ2µ5
µ3
dψ2 + k
H1
F
(
dρ2 + dz2
)
,
24 A. Dimakis and F. Mu¨ller-Hoissen
where
H1 = M0 + b
2M1 + c
2M2 + bcM3 + b
2c2M4,
H2 =
ρ2
µ2µ5
(
µ1µ4
ρ2
M0 − b
2µ4
µ1
M1 − c
2µ1
µ4
M2 − bcM3 + b
2c2
ρ2
µ1µ4
M4
)
,
F = µ23(µ1 − µ4)
2
(
5∏
i=1
Rii
)
R12R214R15R24R
2
25R45
R13R23R34R35
,
ω = −2
(
µ3
µ2µ5
)1/2 (
bR1
√
M0M1 + cR4
√
M0M2 − b
2cR4
√
M1M4 − bc
2R1
√
M2M4
)
,
with
M0 = µ2µ
2
3µ5(µ1 − µ4)
2R212R
2
15R
2
24R
2
45,
M1 = µ
2
1µ
2
2µ3µ
2
5(µ1 − µ3)
2R214R
2
24R
2
45,
M2 = µ
2
2µ3µ
2
4µ
2
5(µ3 − µ4)
2R212R
2
14R
2
15,
M3 = 2µ1µ
2
2µ3µ4µ
2
5(µ1 − µ3)(µ3 − µ4)R11R12R15R24R44R45,
M4 = ρ
4µ21µ
3
2µ
2
4µ
3
5(µ1 − µ3)
2(µ1 − µ4)
2(µ3 − µ4)
2.
With obvious changes in notation, this is the metric obtained in [39].
Remark 4.4. It is plausible that one can start with the diagonal solution of the non-autonomous
chiral model corresponding to a static triple black hole space-time (see (4.1) in [70]) and construct
a space-time with three Myers–Perry black holes. This procedure should continue to produce
solutions with an arbitrary number of rotating black holes.
4.4.4 Bicycling black rings
Let us start with the solution
g˜ = diag
(
−
µ1µ5
µ3µ7
,
ρ2µ3µ7
µ2µ4µ6
,
µ2µ4µ6
µ1µ5
)
of (1.1), which corresponds to a static metric. Removal of a soliton at z = a7 and an anti-soliton
at z = a1 from g˜11, and a rescaling, leads to the seed metric
g0 = w
−1g˜ diag
(
µ27
µ21
, 1, 1
)
= diag
(
µ5
µ3
,−
ρ2µ1µ3
µ2µ4µ6
,−
µ2µ4µ6
µ5µ7
)
, w = −
µ7
µ1
,
with the following solution of (4.1) (up to a constant factor),
f0 =
µ2µ4µ6
µ5µ7
R12R14R16R23R25R27R34R35R36R45R47R56R67√
R11R13R22R224R
2
26R33R44R
2
46R55R57R66
√
R77
.
We have n = 2 and shall set
P˜ = diag
(
−ρ−1µ1, ρµ
−1
7
)
.
The solution of (3.8) is then given by (see Example 3.3)
V =




v11
R13
R15
v12
R12R14R16
µ1R11R13
v13
µ1R15R17
R12R14R16
v21
µ3R57
µ5R37
v22
µ2µ4µ6R17R37
µ1µ3R27R47R67
v23
µ5R27R47R67
µ2µ4µ6R57R77



 .
Binary Darboux Transformations and Vacuum Einstein Equations 25
Without restriction of generality, we can set v11 = v21 = 1. But we do restrict the class of
solutions by setting
v12 = v23 = 0.
Again, the solution Γ of the Stein equation is obtained from (3.18). The resulting solution (3.17)
of the non-autonomous chiral model has to be modified to gˆ = wg, with w as given above, in
order to achieve the determinant condition. We have (disregarding a constant factor)
f(w) = ρ
−1 R17√
R11R77
,
and, from Corollary 4.1 and Proposition 4.2,
fˆ = κρ−5wf0f
−3
(w)
(det P˜ )2 det Γ
det
(
I + P˜
2) ,
which results in a lengthy expression. Setting
v13 = c1, v22 = b2
(a7 − a4)(a7 − a5)(a7 − a6)
(a7 − a1)(a7 − a3)2
, κ = 4(a7 − a1)
2,
we obtain the metric19
ds2 = −
H2
H1
(
dt−
ω1
H2
dϕ−
ω2
H2
dψ
)2
+
1
H2
(
G1dϕ
2 +G2dψ
2 − 2Jdϕdψ
)
+ PH1
(
dρ2 + dz2
)
,
where
H1 = M0 + c
2
1M1 + b
2
2M2 − b
2
2c
2
1M3,
H2 =
µ5
µ3
(
µ1
µ7
M0 − c
2
1
ρ2
µ1µ7
M1 − b
2
2
µ1µ7
ρ2
M2 − b
2
2c
2
1
µ7
µ1
M3
)
,
G1 =
ρ2µ1µ5
µ2µ4µ6
(
M0 − c
2
1
ρ2
µ21
M1 + b
2
2M2 + b
2
2c
2
1
ρ2
µ21
M3
)
,
G2 =
µ2µ4µ6
µ3µ7
(
M0 + c
2
1M1 − b
2
2
µ27
ρ2
M2 + b
2
2c
2
1
µ27
ρ2
M3
)
,
ω1 = b2
R77
µ7
(
µ1µ5
µ2µ4µ6
)1/2(√
M0M2 − c
2
1
ρ
µ1
√
M1M3
)
,
ω2 = c1
R11
µ1
(
µ2µ4µ6
ρ2µ3µ7
)1/2(√
M0M1 − b
2
2
µ7
ρ
√
M2M3
)
,
J = b2c1ρ
2µ1µ2µ3µ4µ
2
5µ6(µ3 − µ7)
2(µ4 − µ7)(µ5 − µ7)(µ6 − µ7)
×R11R12R13R14R
2
15R16R17R27R77,
and
M0 = µ4µ
3
5µ6µ7(µ3 − µ7)
4R212R
2
13R
2
14R
2
16R
2
17R
2
27,
M1 = ρ
2µ21µ2µ3µ
2
4µ5µ
2
6(µ1 − µ7)
2(µ3 − µ7)
4R415R
2
17R
2
27,
19Here we used ai − aj = (µi − µj)(ρ2 + µiµj)/(2µiµj) to eliminate ai − aj .
26 A. Dimakis and F. Mu¨ller-Hoissen
M2 = ρ
4µ1µ2µ
2
3µ
2
5µ7(µ4 − µ7)
2(µ5 − µ7)
2(µ6 − µ7)
2R212R
2
13R
2
14R
2
16,
M3 = ρ
4µ31µ
2
2µ
3
3µ4µ6(µ4 − µ7)
2(µ5 − µ7)
2(µ6 − µ7)
2R415R
2
17,
P =
µ2
µ1µ45µ7(µ3 − µ7)
4
R23R25R34R35R36R45R47R56R57R67
R12R13R14R215R16R17R
2
24R
2
26R27R
2
37R
2
46
.
With obvious changes in notation, this is the “bicycling” black bi-ring solution obtained in [23]
(also see [47]), except for the fact that we have a minus sign instead of a plus in the expressions
for ω1 and ω2.
5 Final remarks
We presented a general formulation of binary Darboux-type transformations in the bidifferential
calculus framework. Whenever a PDDE can be cast into the form (2.2) or (2.3), Theorem 2.1
can be applied and it will typically generate a large class of exact solutions. Meanwhile
a bidifferential calculus formulation is available for quite a number of integrable PDDEs.
We elaborated this general result for the case of the non-autonomous chiral model, consid-
erably extending previous results in [15]. We also presented conditions that, imposed on the
matrix data that determine the general class of solutions, guarantee that the resulting solution
of the non-autonomous chiral model is symmetric (or Hermitian). If the solution is also real,
then it is known to determine a Ricci-flat metric, i.e., a solution of the vacuum Einstein equa-
tions, dimensionally reduced to two dimensions. We essentially solved the equations resulting
from the assumptions in Theorem 2.1 in the case of a diagonal seed metric, though not yet the
V -equations in sufficient generality if P˜ is non-diagonal (but see Examples 4.3 and 4.4). All this
provides a working recipe to compute quite easily solutions of the vacuum Einstein equations.
In particular, in the four-dimensional case we recovered (multi-) Kerr-NUT (in a different way
than in [15]) and the δ = 2 Tomimatsu–Sato solution. In the five-dimensional case we recovered
single and double Myers–Perry black holes, the “black saturn” and the “bicycling black ring”
solutions. The more general solutions still have to be explored. In view of the complexity of the
latter solutions, it is certainly an advantage to have now an independent method at our disposal
to derive, verify or generalize them. Surely further important solutions of Einstein’s equations
in D ≥ 4 space-time dimensions can be recovered using this method and there is a chance to
discover interesting new solutions. We concentrated on examples in the stationary case  = 1,
but developed the formalism as well for the wave case  = −1 [2,4]. It is not difficult to recover
relevant examples in this case too.
The recipe to construct solutions of the non-autonomous chiral model and the dimensionally
reduced vacuum Einstein equations, obtained from Theorem 2.1, is – not surprisingly – a variant,
a sort of matrix version, of the well-known method of Belinski and Zakharov [2–4].
One should look for suitable ways to spot physically relevant solutions within the plethora
of solutions. How are desired properties of solutions, like asymptotic flatness, absence of naked
singularities and proper axis conditions encoded in the (matrix) data that determine a solution?
Here the rod structure analysis [2, 9, 27, 36], developed for the Belinski–Zakharov approach and
frequently used, is of great help.
Section 3 also paved the way toward a treatment of other reductions of the non-autonomous
chiral model, which, e.g., are relevant in the Einstein–Maxwell case and supergravity theories.
In this work we only elaborated Theorem 2.1 for a particular example of an integrable equation
in the bidifferential calculus framework. Although we already applied a more restricted version
of it previously to several other integrable equations, it will be worth to reconsider them and
to also explore further equations, using the much more general solution-generating tool we now
have at our disposal. Furthermore, it should be clarified whether, e.g., the examples in [63–65]
fit into the framework of Theorem 2.1. We should also mention that Sylvester equations, like
Binary Darboux Transformations and Vacuum Einstein Equations 27
those that arise from (2.7), and more generally operator versions of them, are ubiquitous in the
theory of integrable systems. In particular, they are related to a Riemann–Hilbert factorization
problem [66] and they are at the roots of Marchenko’s operator approach [53].
Appendix A. Addendum to Example 4.1
From (4.3) we find that
f[ρ] ∝ ρ1/4, f[wk] ∝ f[w]k
2
, f[w1w2] ∝ f[w1]f[w2]F[w1, w2],
where F[w1, w2] has to solve
(lnF[w1, w2])ρ =
ρ
2
(
(lnw1)ρ(lnw2)ρ − (lnw1)z(lnw2)z
)
,
(lnF[w1, w2])z =
ρ
2
(
(lnw1)ρ(lnw2)z + (lnw1)z(lnw2)ρ
)
(also see [47]). It is easy to verify that
F[w1 · · ·wr, w′1 · · ·w
′
s] ∝
r∏
i=1
s∏
j=1
F[wi, w′j ].
In particular, F[wk1 , w
l
2] ∝ F[w1, w2]
kl. Furthermore, we have
F
[
ρkw1, ρ
lw2
]
∝ ρkl/2wl/21 w
k/2
2 F[w1, w2].
It follows that
f[ρkw] ∝ ρk
2/4wk/2f[w], f[µ˜1 · · · µ˜r] ∝
(
r∏
k=1
f[µ˜k]
)

∏
i<j
F[µ˜i, µ˜j ]

 ,
and
f
[
µ˜1 · · · µ˜r
µ˜′1 · · · µ˜
′
s
]
∝
f[µ˜1 · · · µ˜r]f[µ˜′1 · · · µ˜
′
r]
r∏
i
s∏
j
F[µ˜i, µ˜′j ]
,
from which the main result in Example 4.1 is easily deduced.
Appendix B. Some proofs
Proof of Proposition 4.1. Using (ln detY )ρ = tr(YρY −1) for an invertible (and differentiable)
matrix function Y , we obtain
(ln f)ρ = (ln f0)ρ + tr
(
−ρ−1I + X˜ρX˜
−1
+ P˜ ρP˜
−1
+ Q˜ρQ˜
−1
− P˜ ρP˜
(
1 + P˜
2)−1
− Q˜ρQ˜
(
1 + Q˜
2)−1
)
,
(ln f)z = (ln f0)z + tr
(
X˜zX˜
−1
+ P˜ zP˜
−1
+ Q˜zQ˜
−1
− P˜ zP˜
(
1 + P˜
2)−1
− Q˜zQ˜
(
1 + Q˜
2)−1
)
,
28 A. Dimakis and F. Mu¨ller-Hoissen
for the expression of f in Proposition 4.1. In order to verify that (4.1) holds, we have to show
that these expressions equal the corresponding right hand sides of (4.1), evaluated with
U = −ρφ˜z = −ρ
[
φ˜0,z +
(
UX˜
−1
V
)
z
]
= U0 − ρ
(
U zX˜
−1
V −UX˜
−1
X˜zX˜
−1
V + UX˜
−1
V z
)
,
V = V0 + 
[
ρ
(
UρX˜
−1
V −UX˜
−1
X˜ρX˜
−1
V + UX˜
−1
V ρ
)
+ UX˜
−1
V
]
,
where U0 = −ρφ˜0,z and V0 = (ρφ˜0,ρ + φ˜0).20 We work on the right hand sides of (4.1) and, as
intermediate steps, we consider tr(U2 − U20 ), tr(V
2 − V20 ) and tr(UV − U0V0) separately. First
we eliminate derivatives of φ0 with the help of (3.7) and (3.8). Then we eliminate V U using
the Sylvester equation, and V zU , V ρU via (3.14). Rewriting (3.14) with the help of (3.10), we
obtain a version that allows us to also replace all occurencies of V U z and V Uρ. Several times
one has to exploit the cyclicity of the trace. Finally we use
P˜ ρ = ρ−1P˜
(
I − P˜
2)(
I + P˜
2)−1
, P˜ z = 2ρ−1P˜
2(
I + P˜
2)−1
(which follows from (3.3)), and the corresponding equations for Q˜, to show that (4.1) holds.
Proof of Proposition 4.2. Using Uˆ = U + ρ(lnw)ρI and Vˆ = V + ρ(lnw)zI, we have
tr
(
Uˆ2 − U2
)
= 2ρ(lnw)ρ trU +mρ
2(lnw)2ρ = 2ρ
2(lnw)ρ(ln det g)ρ +mρ
2(lnw)2ρ,
and a corresponding expression for tr(Vˆ2 − V2). Furthermore,
tr
(
Uˆ Vˆ − UV
)
= ρ2
(
(lnw)ρ(ln det g)z + (lnw)z(ln det g)ρ +m(lnw)ρ(lnw)z
)
.
With the help of (4.1) we obtain
(
ln
fˆ
f
)
ρ
=
1
4ρ
tr
(
Uˆ2 − U2 − 
(
Vˆ2 − V2
))
=
ρ
2
(
(lnw)ρ(ln det g)ρ − (lnw)z(ln det g)z
)
+m
ρ
4
(
(lnw)2ρ − (lnw)
2
z
)
,
(
ln
fˆ
f
)
z
=
1
2ρ
tr
(
Uˆ Vˆ − UV
)
=
ρ
2
(
(lnw)ρ(ln det g)z + (lnw)z(ln det g)ρ
)
+m
ρ
2
(lnw)ρ(lnw)z.
Next we use
(ln(ρf(w)))ρ =
ρ
4
(
(lnw)2ρ − (lnw)
2
z
)
, (ln(ρf(w)))z =
ρ
2
(lnw)ρ(lnw)z,
and the corresponding equations for f(det g) and f(w det g), to deduce that
(
ln
f(w det g)
ρf(w)f(det g)
)
ρ
=
ρ
2
(
(lnw)ρ(ln det g)ρ − (lnw)z(ln det g)z
)
,
(
ln
f(w det g)
ρf(w)f(det g)
)
z
=
ρ
2
(
(lnw)ρ(ln det g)z + (lnw)z(ln det g)ρ
)
.
Inserting the last expressions in our previous results, we obtain (4.4) by integration. Let us now
assume that det gˆ = −ρ2, and thus det g = −ρ2w−m. With
f(ρ2wk) ∝ w
k(ρf(w))
k2
(see Example 4.1), (4.5) results from (4.4).
20This is quite a tour de force and a more elegant proof would be desirable.
Binary Darboux Transformations and Vacuum Einstein Equations 29
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