  In this paper, we investigate positive solutions to the following
H\'enon-Sobolev critical system: $$
  -\mathrm{div}(|x|^{-2a}\nabla
u)=|x|^{-bp}|u|^{p-2}u+\nu\alpha|x|^{-bp}|u|^{\alpha-2}|v|^{\beta}u\quad\text{in
}\mathbb{R}^n,$$
  $$ -\mathrm{div}(|x|^{-2a}\nabla
v)=|x|^{-bp}|v|^{p-2}v+\nu\beta|x|^{-bp}|u|^{\alpha}|v|^{\beta-2}v\quad\text{in
}\mathbb{R}^n,$$
  $$u,v\in D_a^{1,2}(\mathbb{R}^n),$$
  where $n\geq 3,-\infty< a<\frac{n-2}{2},a\leq
b<a+1,p=\frac{2n}{n-2+2(b-a)},\nu>0$ and $\alpha>1,\beta>1$ satisfying
$\alpha+\beta=p$. Our findings are divided into two parts, according to the
sign of the parameter $a$.
  For $a\geq 0$, we demonstrate that any positive solution $(u,v)$ is
synchronized, indicating that $u$ and $v$ are constant multiples of positive
solutions to the decoupled H\'enon equation:
  \begin{equation*}
  -\mathrm{div}(|x|^{-2a}\nabla w)=|x|^{-bp}|w|^{p-2}w.
  \end{equation*}
  For $a<0$ and $b>a$, we characterize all nonnegative ground states.
Additionally, we study the nondegeneracy of nonnegative synchronized solutions.
  This work also delves into some general $k$-coupled H\'enon-Sobolev critical
systems.
