Superintegrability and higher order polynomial algebras I

Understand

We present a method to obtain higher order integrals and polynomial algebras for two-dimensional superintegrable systems from creation and annihilation operators.

• All potentials with a second and a third order integrals of motion separable in Cartesian coordinates were studied.
• The integrals of motion of two of them do not generate a cubic algebra.
• We construct for these Hamiltonians a higher order polynomial algebra from the creation and annihilation operators.