Third-order superintegrable systems separable in parabolic coordinates
Mathematical Physics
2015-06-04 v2 math.MP
Exactly Solvable and Integrable Systems
Abstract
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.
Cite
@article{arxiv.1204.0700,
title = {Third-order superintegrable systems separable in parabolic coordinates},
author = {I. Popper and S. Post and P. Winternitz},
journal= {arXiv preprint arXiv:1204.0700},
year = {2015}
}