On superintegrable systems separable in Cartesian coordinates

TL;DR

This paper investigates superintegrable systems separable in Cartesian coordinates, focusing on polynomial integrals of motion related to Chebyshev's theorem, and extends the understanding of their structure and properties.

Contribution

It introduces new superintegrable systems with polynomial integrals of motion connected to Chebyshev's theorem, expanding the class of known separable systems.

Findings

Existence of polynomial integrals of motion linear in angle variables.

Connection between superintegrability and Chebyshev's theorem on binomial differentials.

Extension of superintegrable systems theory in Cartesian coordinates.

Abstract

We continue the study of superintegrable systems of Thompson's type separable in Cartesian coordinates. An additional integral of motion for these systems is the polynomial in momenta of N-th order which is a linear function of angle variables and the polynomial in action variables. Existence of such superintegrable systems is naturally related to the famous Chebyshev theorem on binomial differentials.