Summary: We consider properties of solutions for a class of integral systems in the following: \(u(x)=\int_{\mathbb R^n} \frac{G_\alpha(x-y)v(y)^q}{|y|^\beta} dy\), \(v(x)=\int_{\mathbb R^n} \frac{G_\alpha(x-y)w(y)^r}{|y|^\beta} dy\), \(w(x)=\int_{\mathbb R^n} \frac{G_\alpha(x-y)u(y)^p}{|y|^\beta} dy\), \(x\in \mathbb R^n\), where \(G_\alpha\) is the Bessel potential of order \(\alpha\), \(0\leq \beta<\alpha<n\), \(1<p,q,r<\frac{n-\beta} \beta\), and \(\frac 1{p+1}+\frac q{q+1}>\frac{n-\alpha+\beta}n\), \(\frac 1{q+1}+\frac 1{r+1}>\frac{n-\alpha+\beta} n\), \(\frac 1{r+1}+\frac 1{p+1}>\frac{n-\alpha+\beta} n\). We prove that the positive solutions are symmetric and monotonic by using the moving plane method in integral form.