  We study anti-symmetric solutions about the hyperplane $\{x_n=0\}$ to the
following fractional Hardy-H\'{e}non system $$ \left\{\begin{aligned}
&(-\Delta)^{s_1}u(x)=|x|^\alpha v^p(x),\ \ x\in\mathbb{R}_+^n,
\\&(-\Delta)^{s_2}v(x)=|x|^\beta u^q(x),\ \ x\in\mathbb{R}_+^n, \\&u(x)\geq 0,\
\ v(x)\geq 0,\ \ x\in\mathbb{R}_+^n, \end{aligned}\right. $$ where
$0<s_1,s_2<1$, $n>2\max\{s_1,s_2\}$. Nonexistence of anti-symmetric solutions
are obtained in some appropriate domains of $(p,q)$ under some corresponding
assumptions of $\alpha,\beta$ via the methods of moving spheres and moving
planes. Particularly, for the case $s_1=s_2$, one of our results shows that one
domain of $(p,q)$, where nonexistence of anti-symmetric solutions with
appropriate decay conditions holds true, locates at above the fractional
Sobolev's hyperbola under appropriate condition of $\alpha, \beta$.
