Superintegrable Systems with a Third Order Integrals of Motion

Two-dimensional superintegrable systems with one third order and one lower order integral of motion are reviewed. The fact that Hamiltonian systems with higher order integrals of motion are not the same in classical and quantum mechanics is stressed. New results on the use of classical and quantum third order integrals are presented in Section 5 and 6.

💡 Research Summary

The paper provides a comprehensive review of two‑dimensional superintegrable Hamiltonian systems that possess one third‑order integral of motion together with a lower‑order (first‑ or second‑order) integral. After a brief introduction to the concept of superintegrability—where a system with n degrees of freedom admits more than n independent integrals—the authors recall the well‑known families of superintegrable models that involve only second‑order integrals (e.g., the isotropic harmonic oscillator and the Coulomb potential). They then turn to the central question: under what conditions does a Hamiltonian admit a third‑order integral?

By writing the most general third‑order polynomial in the momenta, \