Separation of Variables and Superintegrability on Riemannian Coverings

Abstract

We introduce Stäckel separable coordinates on the covering manifolds 𝑀ₖ, where 𝑘 is a rational parameter, for certain constant-curvature Riemannian manifolds with a warped manifold structure. These covering manifolds appear implicitly in the literature as connected with superintegrable systems with polynomial momenta first integrals of arbitrarily high degree, such as the Tremblay-Turbiner-Winternitz system. We study here for the first time the multiseparability and superintegrability of natural Hamiltonian systems on these manifolds and see how these properties depend on the parameter 𝑘.