  This paper is focused on the solvability of a family of nonlinear elliptic
systems defined in $\mathbb{R}^N$. Such equations contain Hardy potentials and
Hardy-Sobolev criticalities coupled by a possible critical Hardy-Sobolev term.
That problem arises as a generalization of Gross-Pitaevskii and Bose-Einstein
type systems. By means of variational techniques, we shall find ground and
bound states in terms of the coupling parameter $\nu$ and the order of the
different parameters and exponents. In particular, for a wide range of
parameters we find solutions as minimizers or Mountain-Pass critical points of
the energy functional on the underlying Nehari manifold.
