  It is well known that the Hamiltonian of an $n$-dimensional isotropic
oscillator admits of an $SU(n)$ symmetry, making the system maximally
superintegrable. However, the dynamical symmetries of the anisotropic
oscillator are much more subtle. We introduce a novel set of canonical
transformations that map an $n$-dimensional anisotropic oscillator to the
corresponding isotropic problem. Interestingly, the anisotropic oscillator is
shown to possess the same number of conserved quantities as the isotropic
oscillator, making it maximally superintegrable too. The first integrals are
explicitly calculated in the case of a two-dimensional anisotropic oscillator
and remarkably, they admit closed form expressions.
