  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.
