# Fourth order superintegrable systems separating in Polar Coordinates. I. Exotic Potentials

## TL;DR

This paper classifies all quantum potentials in 2D Euclidean space that separate in polar coordinates and admit a fourth order integral of motion, revealing their connection to Painlevé transcendents and exploring classical analogs.

## Contribution

It provides a complete characterization of exotic superintegrable potentials with fourth order integrals in polar coordinates, linked to Painlevé equations.

## Key findings

- Potentials are expressed via Painlevé P6 transcendent.
- Angular part satisfies a nonlinear ODE with Painlevé property.
- Classical analogs and polynomial algebra of integrals are constructed.

## Abstract

We present all real quantum mechanical potentials in a two-dimensional Euclidean space that have the following properties: 1. They allow separation of variables of the Schr\"odinger equation in polar coordinates, 2. They allow an independent fourth order integral of motion, 3. It turns out that their angular dependent part $S(\theta)$ does not satisfy any linear differential equation. In this case it satisfies a nonlinear ODE that has the Painlev\'e property and its solutions can be expressed in terms of the Painlev\'e transcendent $P_6$. We also study the corresponding classical analogs of these potentials. The polynomial algebra of the integrals of motion is constructed in the classical case.