  By using variational methods, we study the existence of mountain pass
solution to the following doubly critical Schr\"{o}dinger system: $$
  \begin{cases}
  -\Delta u-\mu_1\frac{u}{|x|^2}-|u|^{2^{*}-2}u
&=h(x)\alpha|u|^{\alpha-2}|v|^\beta u\quad \rm{in}\; \R^N,
  -\Delta v-\mu_2\frac{v}{|x|^2}-|v|^{2^{*}-2}v &= h(x)\beta
|u|^{\alpha}|v|^{\beta-2}v\quad \rm{in}\; \R^N,
  \end{cases} $$ where $\alpha\geq 2, \beta\geq 2, \alpha+\beta\leq 2^*$;\; $
\mu_1, \mu_2\in [0, \frac{(N-2)^2}{4})$. The weight function $h(x)$ is allowed
to be sign-changing so that the nonlinearities include a large class of
indefinite weights. We show that the $PS$ condition is satisfied at higher
energy level when $\alpha+\beta=2^*$ and obtain the existence of mountain pass
solution. Besides, a nonexistence result of the ground state is given.
