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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 certainfunction defined on the configuration space and depending of a parameter$\kappa$, is studied. The new Hamiltonians, and the associated constants ofmotion $J_{ri}$, $i=1,2,3$, 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 isdevoted to the existence of higher-order superintegrability. The resultsobtained in the second part are related with the TTW and the PW systems.

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