%0 Journal Article
%T Third-order superintegrable systems separable in parabolic coordinates
%A I. Popper
%A S. Post
%A P. Winternitz
%J Physics
%D 2012
%I arXiv
%R 10.1063/1.4729248
%X 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.
%U http://arxiv.org/abs/1204.0700v2