# Higher Order Quantum Superintegrability: a new "Painlev\'e conjecture"

## TL;DR

This paper reviews superintegrable quantum systems in 2D Euclidean space, highlighting new nonlinear equations with Painlevé property for higher-order integrals and proposing a conjecture for all N≥3.

## Contribution

It introduces a new conjecture that nonlinear equations governing exotic potentials possess the Painlevé property for all N≥3, expanding understanding of superintegrable quantum systems.

## Key findings

- Existence of two types of superintegrable potentials: standard and exotic.
- For N=3, 4, 5, the governing equations have the Painlevé property.
- A polynomial algebra of integrals of motion is constructed for these systems.

## Abstract

We review recent results on superintegrable quantum systems in a two-dimensional Euclidean space with the following properties. They are integrable because they allow the separation of variables in Cartesian coordinates and hence allow a specific integral of motion that is a second order polynomial in the momenta. Moreover, they are superintegrable because they allow an additional integral of order $N>2$. Two types of such superintegrable potentials exist. The first type consists of "standard potentials" that satisfy linear differential equations. The second type consists of "exotic potentials" that satisfy nonlinear equations. For $N= 3$, 4 and 5 these equations have the Painlev\'e property. We conjecture that this is true for all $N\geq3$. The two integrals X and Y commute with the Hamiltonian, but not with each other. Together they generate a polynomial algebra (for any $N$) of integrals of motion. We show how this algebra can be used to calculate the energy spectrum and the wave functions.