Abstract:
We review systems in $E(2)$ that are separable in Cartesian coordinates and admit a third-order integral both in quantum mechanics and in classical mechanics. Differences and similarities between those two cases are illustrated by numerous examples. Many of these superintegrable systems are new, and a relation is seen between superintegrable potentials and Painlevé transcendents.
Keywords:integrals of motion, superintegrability, separation of variables.