Superintegrable systems in non-Euclidean plane: hidden symmetries leading to linearity

TL;DR

This paper demonstrates that nineteen classical superintegrable systems in two-dimensional non-Euclidean spaces have hidden symmetries that enable their linearization, revealing new structural insights into these complex systems.

Contribution

It identifies hidden symmetries in nineteen superintegrable systems in non-Euclidean spaces, leading to their linearization, which is a novel unifying approach for these systems.

Findings

Hidden symmetries enable linearization of systems.

Nineteen specific superintegrable systems analyzed.

Results unify understanding of non-Euclidean superintegrable systems.

Abstract

Nineteen classical superintegrable systems in two-dimensional non-Euclidean spaces are shown to possess hidden symmetries leading to their linearization. They are the two Perlick systems [A. Ballesteros, A. Enciso, F.J. Herranz and O. Ragnisco, Class. Quantum Grav. 25, 165005 (2008)], the Taub-NUT system [A. Ballesteros, A. Enciso, F.J. Herranz, O. Ragnisco, and D. Riglioni, SIGMA 7, 048 (2011)], and all the seventeen superintegrable systems for the four types of Darboux spaces as determined in [E.G. Kalnins, J.M. Kress, W. Miller, P. Winternitz, J. Math. Phys. 44, 5811--5848 (2003)].