Third-order superintegrable systems separable in parabolic coordinates

Details Third-order superintegrable systems separable in parabolic coordinates Third-order superintegrable systems separable in parabolic coordinates

2012-04-03

arxiv.org/abs/1204.0700v2

Physics

Mathematical Physics

Scientific paper

In this paper, we investigate superintegrable systems which separate in parabolic coordinates and admit a third-order integral of motion. We give the corresponding determining equations and show that all such systems are multi-separable and so admit two second-order integrals. The third-order integral is their Lie or Poisson commutator. We discuss how this situation is different from the Cartesian and polar cases where new potentials were discovered which are not multi-separable and which are expressed in terms of Painlev\'e transcendents or elliptic functions.

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