In this paper (arXiv:1403.1829) we present a new integrable generalization to the 2D sphere and to the hyperbolic space of the 2D Euclidean anisotropic oscillator Hamiltonian with Rosochatius (centrifugal) terms, and its curved integral of the motion is shown to be quadratic in the momenta. In order to construct such a new integrable Hamiltonian, we  make use of a group theoretical approach in which the curvature of the underlying space is treated as an additional (contraction) parameter, and we make extensive use of projective coordinates and their associated phase spaces. These findings supports 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.