Separation of Variables and Superintegrability on Riemannian Coverings

TL;DR

This paper explores how separation of variables and superintegrability properties of Hamiltonian systems are affected by the parameter on Riemannian coverings of constant-curvature manifolds, linking geometric structures to integrable systems.

Contribution

It introduces Stäckel separable coordinates on Riemannian coverings and analyzes their superintegrability, revealing dependence on the parameter $k$ for the first time.

Findings

Identification of Stäckel separable coordinates on covering manifolds.

Analysis of superintegrability dependence on the parameter $k$.

Connection to known superintegrable systems like the Tremblay-Turbiner-Winternitz system.

Abstract

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