**Abstract:**
Two infinite families of exactly solvable and superintegrable
potentials on a
plane are introduced. Both depend on an integer or rational parameter k
and allow the separation of variables in polar coordinates. One is a
deformation of the isotropic harmonic oscillator, the other of the
Coulomb- Kepler potential. The underlying algebraic structure of the new
potentials is revealed as well as their hidden algebra. In the quantum
case we
calculate the wave functions and energy spectrum with its degeneracy. In
the classical one we calculate the bounded trajectories and show that they
are all periodic, as required for a
maximally superintegrable system.