A new integrable anisotropic oscillator on the two-dimensional sphere and the hyperbolic plane

A new integrable generalization to the 2D sphere $S^2$ and to the hyperbolic space $H^2$ of the 2D Euclidean anisotropic oscillator Hamiltonian with Rosochatius (centrifugal) terms is presented, and its curved integral of the motion is shown to be quadratic in the momenta. In order to construct such a new integrable Hamiltonian $H_\kappa$, we will make use of a group theoretical approach in which the curvature $\kappa$ of the underlying space will be treated as an additional (contraction) parameter, and we will make extensive use of projective coordinates and their associated phase spaces. It turns out that when the oscillator parameters $\Omega_1$ and $\Omega_2$ are such that $\Omega_2=4\Omega_1$, the system turns out to be the well-known superintegrable 1:2 oscillator on $S^2$ and $H^2$. Nevertheless, numerical integration of the trajectories of $H_\kappa$ suggests that for other values of the parameters $\Omega_1$ and $\Omega_2$ the system is not superintegrable. In this way, we support the conjecture that for each commensurate (and thus superintegrable) $m:n$ Euclidean oscillator there exists a two-parametric family of curved integrable (but not superintegrable) oscillators that turns out to be superintegrable only when the parameters are tuned to the $m:n$ commensurability condition.