  The paper is concerned with the existence and asymptotic properties of
normalized ground states of the following nonlinear Schr\"odinger system with
critical exponent: \begin{equation*}
  \left\{\begin{aligned}
  &-\delta u+\lambda_1 u=|u|^{2^*-2}u+{\nu\alpha} |u|^{\alpha-2}|v|^\beta
u,\quad \text{in }\mathbb{R}^N,
  &-\delta v+\lambda_2 v=|v|^{2^*-2}v+{\nu\beta} |u|^\alpha
|v|^{\beta-2}v,\quad \text{in }\mathbb{R}^N,
  &\int u^2=a^2,\;\;\; \int v^2=b^2,
  \end{aligned} \right. \end{equation*} where $N=3,4$, $\alpha,\beta>1$,
$2<\alpha+\beta<2^*=\frac{2N}{N-2}$. We prove that a normalized ground state
does not exist for $\nu<0$. When $\nu>0$ and $\alpha+\beta\le 2+\frac{4}{N}$,
we show that the system has a normalized ground state solution for
$0<\nu<\nu_0$, the constant $\nu_0$ will be explicitly given. In the case
$\alpha+\beta>2+\frac{4}{N}$ we prove the existence of a threshold $\nu_1\ge 0$
such that a normalized ground state solution exists for $\nu>\nu_1$, and does
not exist for $\nu<\nu_1$. We also give conditions for $\nu_1=0$. Finally we
obtain the asymptotic behavior of the minimizers as $\nu\to0^+$ or
$\nu\to+\infty$.
