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 $\lambda$ modifies the Hamilton–Jacobi equation but preserves the existence of a multiple separability.