$N^{th}$-Order Superintegrable Systems Separating in Polar Coordinates

Abstract

Classical and quantum Hamiltonian systems in two-dimensional Euclidean plane and allowing separation of variables in polar coordinates are investigated. The additional integral of motion is assumed to be a polynomial of degree $N \geq 3$ in momenta. After analyzing the particular cases of $N = 3, 4$ and $5$, a general description will be given. This leads to a classification of superintegrable potentials into two major categories. For the exotic potentials, the existence of an infinite family of superintegrable potentials in terms of the sixth Painlevé transcendent $P_6$ is conjectured and will be demonstrated for the first few cases.