Third-order superintegrable systems separable in parabolic coordinates

TL;DR

This paper classifies third-order superintegrable systems separable in parabolic coordinates, showing they are multi-separable with two second-order integrals, contrasting with other coordinate systems where more complex potentials arise.

Contribution

It provides a complete characterization of third-order superintegrable systems in parabolic coordinates and highlights their multi-separable nature, differing from Cartesian and polar cases.

Findings

All such systems are multi-separable with two second-order integrals.

The third-order integral is derived from the Lie or Poisson commutator of the second-order integrals.

Contrasts with Cartesian and polar cases where non-multi-separable potentials involve Painlevé transcendents or elliptic functions.

Abstract

In this paper, we investigate superintegrable systems which separate in parabolic coordinates and admit a third-order integral of motion. We give the corresponding determining equations and show that all such systems are multi-separable and so admit two second-order integrals. The third-order integral is their Lie or Poisson commutator. We discuss how this situation is different from the Cartesian and polar cases where new potentials were discovered which are not multi-separable and which are expressed in terms of Painlev\'e transcendents or elliptic functions.