We study the system involving fully nonlinear nonlocal operators:
$$\begin{array}{cc}\hfill {F}_{\alpha}\left(u\left(x\right)\right)& ={C}_{n,\alpha}PV{\int}_{{\mathbb{R}}^{n}}\frac{G\left(u\left(x\right)u\left(y\right)\right)}{xy{}^{n+\alpha}}\phantom{\rule{0.3em}{0ex}}dy=f\left(u\left(x\right),v\left(x\right)\right),\hfill \\ \hfill {F}_{\beta}\left(v\left(x\right)\right)& ={C}_{n,\beta}PV{\int}_{{\mathbb{R}}^{n}}\frac{G\left(v\left(x\right)v\left(y\right)\right)}{xy{}^{n+\beta}}\phantom{\rule{0.3em}{0ex}}dy=g\left(u\left(x\right),v\left(x\right)\right).\hfill \end{array}$$
We will prove the symmetry and monotonicity for positive solutions to the nonlinear
system in whole space by using the method of moving planes. To achieve it, a narrow
region principle and a decay at infinity are established. Further more, nonexistence of
positive solutions to the nonlinear system on a half space is derived. In addition, the
symmetry and monotonicity in whole space for positive solutions to a fully nonlinear
nonlocal system
$${F}_{\alpha}\left(u\left(x\right)\right)={u}^{p}\left(x\right)+{v}^{q}\left(x\right),\phantom{\rule{1em}{0ex}}{F}_{\beta}\left(v\left(x\right)\right)={v}^{p}\left(x\right)+{u}^{q}\left(x\right)$$
can be derived.
