Third-order superintegrable systems separable in parabolic coordinates

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 Painlevé transcendents or elliptic functions.

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• Superintegrable systems with spin and second-order integrals of motion
• A new approach to the higher-order superintegrability of the Tremblay-Turbiner-Winternitz system
• Fourth-order superintegrable systems separating in Polar Coordinates. II. Standard Potentials
• Higher Order Quantum Superintegrability: a new "Painlevé conjecture"
• Two-dimensional superintegrable systems from operator algebras in one dimension
• Superintegrable systems with spin and second-order (pseudo)tensor integrals of motion
• (Super-)integrable systems associated to 2-dimensional projective connections with one projective symmetry
• Higher order superintegrability, Painlevé transcendents and representations of polynomial algebras
• A family of fourth-order superintegable systems with rational potentials related to Painlevé VI
• Cubic first integrals of autonomous dynamical systems in $E^2$ by an algorithmic approach
• Polynomially Superintegrable Hamiltonians Separating in Cartesian Coordinates