Third order superintegrable systems separating in polar coordinates

A complete classification is presented of quantum and classical superintegrable systems in $E_2$ that allow the separation of variables in polar coordinates and admit an additional integral of motion of order three in the momentum. New quantum superintegrable systems are discovered for which the potential is expressed in terms of the sixth Painlev'e transcendent or in terms of the Weierstrass elliptic function.

💡 Research Summary

The paper undertakes a comprehensive classification of both quantum and classical superintegrable systems defined on the two‑dimensional Euclidean plane (E_{2}) that admit separation of variables in polar coordinates and possess an additional integral of motion that is cubic in the momenta. Superintegrability, by definition, requires a system with (n) degrees of freedom to have (2n-1) independent integrals; in two dimensions this means three mutually commuting (or, in the quantum case, mutually compatible) operators. While many works have catalogued systems with first‑ or second‑order integrals, the landscape of third‑order integrals has remained largely unexplored, especially under the additional constraint of polar separability.

The authors start by assuming a Hamiltonian of the form
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