Polynomial Poisson algebras for classical superintegrable systems with a third-order integral of motion

Abstract

We present polynomial Poisson algebras for the eight classical potentials in two-dimensional Euclidian space that separate in Cartesian coordinates and allow a third-order integral of motion. Some of the classical superintegrable potentials do not coincide with quantum ones, but are their singular limits. We show that all bounded trajectories in these potentials are periodic.