Dynamical symmetries of the anisotropic oscillator

It is well known that the Hamiltonian of an $n$-dimensional isotropic oscillator admits 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. Consequently, the anisotropic oscillator is found to possess the same number of conserved quantities as the isotropic oscillator, making it maximally superintegrable too (commensurate case). The first integrals are explicitly calculated in the case of a two-dimensional anisotropic oscillator and remarkably, they admit closed-form expressions.