<div>In this paper, we study the problem for a nonlinear elliptic system
involving fractional Laplacion: $$ \begin{cases}
\
(-\Delta)^{\frac{\alpha}{2}}u=|x|^{\gamma}u^{p}v^{q+1},\\
\
(-\Delta)^{\frac{\beta}{2}}v=|x|^{\tau}u^{p+1}v^{q},
\end{cases} $$ where
$0&lt;\alpha,\beta&lt;2,$
$p,q&gt;0\ and\
\max\{p,q\}\geq1,$
$\alpha+\gamma&gt;0,\beta+\tau&gt;0,$
$n\geq2$. First of all, while in the subcritical case,
i.e.
$n+\alpha+\gamma-p(n-\alpha)-(q+1)(n-\beta)&gt;0$,
$n+\beta+\tau-(p+1)(n-\alpha)-q(n-\beta)&gt;0$,
we prove the nonexistence of positive solution for the above system in
$\mathbb{R}^{n}$. Moreover, though
$Doubling\ Lemma$ to obtain the singularity estimates
of the positive solution on bounded domain $\Omega$. In
addition, while in the critical case, i.e.
$n+\alpha+\gamma-p(n-\alpha)-(q+1)(n-\beta)=0$,
$n+\beta+\tau-(p+1)(n-\alpha)-q(n-\beta)=0$,
we show that the positive solution of above system are radical symmetric
and decreasing about some point by using the method of
$Moving\ planes$ in
$\mathbb{R}^{n}$.</div>