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In the present paper, we study the normalized solutions with least energy tothe following system: $$\begin{cases} -\Delta u+\lambda_1u=\mu_1|u|^{p-2}u+\beta r_1|u|^{r_1-2}|v|^{r_2}u\quad &\hbox{in}\;\mathbb R^N,\\-\Delta v+\lambda_2v=\mu_2 |v|^{q-2}v+\betar_2|u|^{r_1}|v|^{r_2-2}v\quad&\hbox{in}\;\mathbb R^N,\\ \int_{\mathbbR^N}u^2=a_1^2\quad\hbox{and}\;\int_{\mathbb R^N}v^2=a_2^2, \end{cases}$$ where$p,q,r_1+r_2$ can be Sobolev critical. To this purpose, we study the geometryof the Pohozaev manifold and the associated minimizition problem. Under someassumption on $a_1,a_2$ and $\beta$, we obtain the existence of the positivenormalized ground state solution to the above system. We have solved someunsolved open problems in this area.

Normalized ground states for semilinear elliptic systems with critical and subcritical nonlinearities