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

Dynamical symmetries of the anisotropic oscillator