  A nonlinear two-dimensional system is studied by making use of both the
Lagrangian and the Hamiltonian formalisms. The present model is obtained as a
two-dimensional version of a one-dimensional oscillator previously studied at
the classical and also at the quantum level. First, it is proved that it is a
super-integrable system, and then the nonlinear equations are solved and the
solutions are explicitly obtained. All the bounded motions are quasiperiodic
oscillations and the unbounded (scattering) motions are represented by
hyperbolic functions. In the second part the system is generalized to the case
of $n$ degrees of freedom. Finally, the relation of this nonlinear system with
the harmonic oscillator on spaces of constant curvature, two-dimensional sphere
$S^2$ and hyperbolic plane $H^2$, is discussed.
