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An integrable generalization on the two-dimensional sphere S^2 and thehyperbolic plane H^2 of the Euclidean anisotropic oscillator Hamiltonian with"centrifugal" terms given by $H=1/2(p_1^2+p_2^2)+ \delta q_1^2+(\delta +\Omega)q_2^2 +\frac{\lambda_1}{q_1^2}+\frac{\lambda_2}{q_2^2}$ is presented.The resulting generalized Hamiltonian H_\kappa\ depends explicitly on theconstant Gaussian curvature \kappa\ of the underlying space, in such a way thatall the results here presented hold simultaneously for S^2 (\kappa>0), H^2(\kappa<0) and E^2 (\kappa=0). Moreover, H_\kappa\ is explicitly shown to beintegrable for any values of the parameters \delta, \Omega, \lambda_1 and\lambda_2. Therefore, H_\kappa\ can also be interpreted as an anisotropicgeneralization of the curved Higgs oscillator, that is recovered as theisotropic limit \Omega=0 of H_\kappa. Furthermore, numerical integration ofsome of the trajectories for H_\kappa\ are worked out and the dynamicalfeatures arising from the introduction of a curved background are highlighted.  The superintegrability issue for H_\kappa\ is discussed by focusing on thevalue \Omega=3\delta, which is one of the cases for which the EuclideanHamiltonian H is known to be superintegrable (the 1:2 oscillator). We shownumerically that for \Omega=3\delta\ the curved Hamiltonian H_\kappa\ presentsnonperiodic bounded trajectories, which seems to indicate that H_\kappa\provides a non-superintegrable generalization of H. We compare this result witha previously known superintegrable curved analogue H'_\kappa\ of the 1:2Euclidean oscillator showing that the \Omega=3\delta\ specialization ofH_\kappa\ does not coincide with H'_\kappa. Finally, the geometricalinterpretation of the curved "centrifugal" terms appearing in H_\kappa\ is alsodiscussed in detail.

The anisotropic oscillator on the two-dimensional sphere and the hyperbolic plane