  We deform N-dimensional (Euclidean, spherical and hyperbolic) oscillator and
Coulomb systems, replacing their angular degrees of freedom by those of a
generalized rational Calogero model. Using the action-angle description, it is
established that maximal superintegrability is retained. For the rational
Calogero model with Coulomb potential, we present all constants of motion via
matrix model reduction. In particular, we construct the analog of the
Runge-Lenz vector.
