  In the present paper, we study the normalized solutions with least energy to
the following system: $$\begin{cases} -\Delta u+\lambda_1u=\mu_1
|u|^{p-2}u+\beta r_1|u|^{r_1-2}|v|^{r_2}u\quad &\hbox{in}\;\mathbb R^N,\\
-\Delta v+\lambda_2v=\mu_2 |v|^{q-2}v+\beta
r_2|u|^{r_1}|v|^{r_2-2}v\quad&\hbox{in}\;\mathbb R^N,\\ \int_{\mathbb
R^N}u^2=a_1^2\quad\hbox{and}\;\int_{\mathbb R^N}v^2=a_2^2, \end{cases}$$ where
$p,q,r_1+r_2$ can be Sobolev critical. To this purpose, we study the geometry
of the Pohozaev manifold and the associated minimizition problem. Under some
assumption on $a_1,a_2$ and $\beta$, we obtain the existence of the positive
normalized ground state solution to the above system. We have solved some
unsolved open problems in this area.
