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

The superintegrability of four Hamiltonians $\tilde{H_r} = \lambda\, H_r$, $r=a,b,c,d$, where $H_r$ are known Hamiltonians and $\lambda$ is a certain function defined on the configuration space and depending of a parameter $\kappa$, is studied. The new Hamiltonians, and the associated constants of motion $J_{ri}$, $i=1,2,3$, are continous functions of the parameter $\kappa$. 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 TTW and the PW systems.