Wednesday, April 4, 2012

1204.0700 (I. Popperi et al.)

Third-order superintegrable systems separable in parabolic coordinates    [PDF]

I. Popperi, S. Post, P. Winternitz
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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