Manifold Ways to Darboux-Halphen System

Abstract

Many distinct problems give birth to the Darboux-Halphen system of differential equations, and here we review some of them. The first is the classical problem presented by Darboux and later solved by Halphen concerning finding an infinite number of double orthogonal surfaces in R³. The second is a problem in general relativity about a gravitational instanton in the Bianchi IX metric space. The third problem stems from the new take on the moduli of enhanced elliptic curves called the Gauss-Manin connection in disguise, developed by one of the authors, and finally, in the last problem Darboux-Halphen system emerges from the associative algebra on the tangent space of a Frobenius manifold.