A nonlinear deformation of the isotonic oscillator and the Smorodinski-Winternitz system:: Integrability and superintegrability

The properties of a nonlinear deformation of the isotonic oscillator are studied. This deformation affects to both the kinetic term and the potential and depends on a parameter lambda in such a way that for lambda = 0 all the characteristics of of the classical system are recovered. In the second part, that is devoted to the two-dimensional case, a lambda-dependent deformation of the Smorodinski-Winternitz system is studied. It is proved that the deformation introduced by the parameter A modifies the Hamilton-Jacobi equation but preserves the existence of a multiple separability.