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

Abstract

The higher order superintegrability of systems separable in polar coordinates is studied using an approach 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.