# Fourth order Superintegrable systems separating in Cartesian coordinates I. Exotic quantum potentials

## TL;DR

This paper classifies exotic quantum potentials in 2D superintegrable systems with fourth order integrals, showing they satisfy nonlinear ODEs with the Painlevé property and can be integrated using special functions.

## Contribution

It identifies all quantum potentials that do not satisfy linear differential equations but are solvable via Painlevé transcendents or elliptic functions.

## Key findings

- All such potentials satisfy nonlinear ODEs with Painlevé property.
- Solutions are expressed in terms of Painlevé transcendents or elliptic functions.
- Provides a complete classification of these exotic quantum potentials.

## Abstract

A study is presented of two-dimensional superintegrable systems separating in Cartesian coordinates and allowing an integral of motion that is a fourth order polynomial in the momenta. All quantum mechanical potentials that do not satisfy any linear differential equation are found. They do however satisfy nonlinear ODEs. We show that these equations always have the Painlev\'e property and integrate them in terms of known Painlev\'e transcendents or elliptic functions.