Superintegrable deformations of superintegrable systems: Quadratic superintegrability and higher-order superintegrability

The superintegrability of four Hamiltonians $H r ˜ =λ H r$⁠, r = a, b, c, d, where H_r are known Hamiltonians and λ is a certain function defined on the configuration space and depended on a parameter κ, is studied. The new Hamiltonians, and the associated constants of motion J_ri, i = 1, 2, 3, are continous functions of the parameter κ. The first part is concerned with separability and quadratic superintegrability (the integrals of motion are quadratic in the momenta) and the second part is devoted to the existence of higher-order superintegrability. The results obtained in the second part are related with the Tremblay-Turbiner-Winternitz and the Post–Winternitz systems.