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

A new proof of the higher-order superintegrability of a noncentral oscillator with inversely quadratic nonlinearities