We study the existence of bound and ground states for a class of nonlinear elliptic systems in $mathbb{R}^N$. These equations involve critical power nonlinearities and Hardy-type singular potentials, coupled by a term containing up to critical powers. More precisely, we find ground states either the positive coupling parameter $ u$ is large or $ u$ is small under suitable assumptions on the other parameters of the problem. Furthermore, bound states are found as Mountain-Pass-type critical points of the underlying functional constrained on the Nehari manifold. Our variational approach improves some known results and allows us to cover range of parameters which have not been considered previously.
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