We consider the following Hartree system with Hardy term \(\begin{aligned} {\left\{ \begin{array}{ll} \displaystyle -\Delta u-\gamma _1\frac{u}{|x|^2}=(|x|^{-\mu }*|u|^{2^*_\mu })|u|^{2^*_\mu -2}u+\omega (|x|^{-\mu }*|v|^{2^*_\mu })|u|^{2^*_\mu -2}u\quad \text {in}~\mathbb {R}^N,\\ \displaystyle -\Delta v-\gamma _2\frac{v}{|x|^2}=(|x|^{-\mu }*|v|^{2^*_\mu })|v|^{2^*_\mu -2}v+\omega (|x|^{-\mu }*|u|^{2^*_\mu })|v|^{2^*_\mu -2}v~\quad \text {in}~\mathbb {R}^N,\\ \end{array}\right. } \end{aligned}\) where \(N\ge 3\) , \(2^*_\mu :=\frac{2N-\mu }{N-2}\) , \(0<\mu <N\) , \(\omega >0\) , \(0\le \gamma _1,\gamma _{2}<C^*:=(\frac{N-2}{2})^2\) and \(C^*\) is the Hardy constant. For this system, we first establish the existence of positive ground state solutions by variational methods. Additionally, the moving-plane method and the Kelvin transform are applied to establish the symmetry of the ground state solutions if \(0<\mu <\min \{4,N\}\) . Finally, by proving the symmetry and regularity of the solutions, we can study the asymptotic behavior of the solutions at the origin and at infinity.
