General $N^{th}$-order superintegrable systems separating in polar coordinates

TL;DR

This paper classifies superintegrable systems in 2D Euclidean space with polynomial integrals of order N, exploring their properties and proposing a new family involving the sixth Painlevé transcendent.

Contribution

It provides a general description of Nth-order superintegrable systems in polar coordinates and introduces a conjecture about an infinite family related to Painlevé transcendents.

Findings

Potentials classified into two main classes

Demonstration of a new family involving Painlevé P6

Conjecture of an infinite family of superintegrable potentials

Abstract

The general description of superintegrable systems with one polynomial integral of order $N$ in the momenta is presented for a Hamiltonian system in two-dimensional Euclidean plane. We consider classical and quantum Hamiltonian systems allowing separation of variables in polar coordinates. The potentials can be classified into two major classes and their main properties are described. We conjecture that a new infinite family of superintegrable potentials in terms of the sixth Painlev\'e transcendent $P_6$ exists and demonstrate this for the first few cases.