Hamiltonians separable in cartesian coordinates and third-order integrals of motion

TL;DR

This paper classifies Hamiltonian systems in two-dimensional Euclidean space that are separable in Cartesian coordinates and possess third-order integrals of motion, revealing new superintegrable systems and connections to integrable equations.

Contribution

It provides a complete classification of such Hamiltonian systems, including many new superintegrable potentials, and uncovers links to integrable equations like Korteweg-De Vries and Painlevé transcendents.

Findings

Many new superintegrable systems identified

Connections established between quantum potentials and integrable equations

Classification includes both quantum and classical Hamiltonian systems

Abstract

We present in this article all Hamiltonian systems in E(2) that are separable in cartesian coordinates and that admit a third-order integral, both in quantum and in classical mechanics. Many of these superintegrable systems are new, and it is seen that there exists a relation between quantum superintegrable potentials, invariant solutions of the Korteweg-De Vries equation and the Painlev\'e transcendents.