Third-order superintegrable systems separable in parabolic coordinates
  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. 
I. Popper, S. Post and P. Winternitz
math-ph math.MP nlin.SI