  The superintegrability of a rational harmonic oscillator (non-central
harmonic oscillator with rational ratio of frequencies) with non-linear
"centrifugal" terms is studied. In the first part, the system is directly
studied in the Euclidean plane; the existence of higher-order
superintegrability (integrals of motion of higher order than 2 in the momenta)
is proved by introducing a deformation in the quadratic complex equation of the
linear system. The constants of motion of the nonlinear system are explicitly
obtained. In the second part, the inverse problem is analyzed in the general
case of $n$ degrees of freedom; starting with a general Hamiltonian $H$, and
introducing appropriate conditions for obtaining superintegrability, the
particular "centrifugal" nonlinearities are obtained.
