Third-order superintegrable systems with potentials satisfying nonlinear equations

TL;DR

This paper investigates two-dimensional superintegrable systems with third-order integrals, revealing that only Cartesian and polar coordinate systems admit potentials described by nonlinear special functions, while others have potentials with fixed singularities.

Contribution

It classifies superintegrable systems based on coordinate separation and characterizes the nature of their potentials, highlighting the special role of Cartesian and polar coordinates.

Findings

Only Cartesian and polar coordinates admit potentials with nonlinear special functions.

Parabolic and elliptic coordinates have potentials with non-movable singularities.

The study advances understanding of the structure of superintegrable systems with third-order integrals.

Abstract

The conditions for superintegrable systems in two-dimensional Euclidean space admitting separation of variables in an orthogonal coordinate system and a functionally independent third-order integral are studied. It is shown that only systems that separate in subgroup type coordinates, Cartesian or polar, admit potentials that can be described in terms of nonlinear special functions. Systems separating in parabolic or elliptic coordinated are shown to have potentials with only non-movable singularities.