A new approach to the higher-order superintegrability of the Tremblay-Turbiner-Winternitz system

TL;DR

This paper introduces a novel method for analyzing higher-order superintegrability in polar coordinate systems, successfully proving superintegrability of the TTW system and deriving explicit constants of motion.

Contribution

It presents a new approach using factorization of constants of motion, extending previous techniques to the TTW system and related systems.

Findings

Proves superintegrability of the TTW system.

Provides explicit expressions for constants of motion.

Identifies a second related family of superintegrable systems.

Abstract

The higher-order superintegrability of systems separable in polar coordinates is studied using an approch that was previously applied for the study of the superintegrability of a generalized Smorodinsky-Winternitz system. The idea is that the additional constant of motion can be factorized as the product of powers of two particular rather simple complex functions (here denoted by $M$ and $N$). This technique leads to a proof of the superintegrability of the Tremblay-Turbiner-Winternitz system and to the explicit expression of the constants of motion. A second family (related with the first one) of superintegrable systems is also studied.