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

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 Painlevé transcendent $P_6$ exists and demonstrate this for the first few cases.

arXiv comment17 pages

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• Superintegrable systems with spin and second-order (pseudo)tensor integrals of motion
• New superintegrable models on spaces of constant curvature
• A fourth-order superintegrable system with a rational potential related to Painleve VI
• On higher-dimensional superintegrable systems: A new family of classical and quantum Hamiltonian models
• A family of fourth-order superintegable systems with rational potentials related to Painlevé VI
• Superintegrability and Deformed Oscillator Realizations of Quantum TTW Hamiltonians on Constant-Curvature Manifolds and with Reflections in a Plane
• Doubly Exotic $N$th-order Order Superintegrable Classical Systems Separating in Cartesian Coordinates
• Integrable systems in magnetic fields: the generalized parabolic cylindrical case
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• More on superintegrable models on spaces of constant curvature
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