Are all classical superintegrable systems in two-dimensional space linearizable?

TL;DR

This paper investigates whether all classical superintegrable systems in two-dimensional space can be transformed into linear systems, highlighting known examples with hidden symmetries and proposing a conjecture for all such systems.

Contribution

It demonstrates that certain classical superintegrable systems possess hidden symmetries enabling linearization and conjectures this applies universally to all such systems.

Findings

Examples show hidden symmetries lead to linearization.

The Tremblay-Turbiner-Winternitz system is linearizable.

Conjecture that all 2D superintegrable systems have hidden symmetries.

Abstract

Several examples of classical superintegrable systems in two-dimensional spac are shown to possess hidden symmetries leading to their linearization. They are those determined 50 years ago in [Phys. Lett. 13, 354 (1965)], and the more recent Tremblay-Turbiner-Winternitz system [J. Phys. A: Math. Theor. 42, 242001 (2009)]. We conjecture that all classical superintegrable systems in two-dimensional space have hidden symmetries that make them linearizable.