  An integrable generalization on the two-dimensional sphere S^2 and the
hyperbolic 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 the
constant Gaussian curvature \kappa\ of the underlying space, in such a way that
all 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 be
integrable for any values of the parameters \delta, \Omega, \lambda_1 and
\lambda_2. Therefore, H_\kappa\ can also be interpreted as an anisotropic
generalization of the curved Higgs oscillator, that is recovered as the
isotropic limit \Omega=0 of H_\kappa. Furthermore, numerical integration of
some of the trajectories for H_\kappa\ are worked out and the dynamical
features arising from the introduction of a curved background are highlighted.
  The superintegrability issue for H_\kappa\ is discussed by focusing on the
value \Omega=3\delta, which is one of the cases for which the Euclidean
Hamiltonian H is known to be superintegrable (the 1:2 oscillator). We show
numerically that for \Omega=3\delta\ the curved Hamiltonian H_\kappa\ presents
nonperiodic bounded trajectories, which seems to indicate that H_\kappa\
provides a non-superintegrable generalization of H. We compare this result with
a previously known superintegrable curved analogue H'_\kappa\ of the 1:2
Euclidean oscillator showing that the \Omega=3\delta\ specialization of
H_\kappa\ does not coincide with H'_\kappa. Finally, the geometrical
interpretation of the curved "centrifugal" terms appearing in H_\kappa\ is also
discussed in detail.
