  We prove the existence of infinitely many non-radial positive solutions for
the Schr\"{o}dinger-Newton system $$
  \left\{\begin{array}{ll}
  \Delta u- V(|x|)u + \Psi u=0, &x\in\mathbb{R}^3,\newline
  \Delta \Psi+\frac12 u^2=0, &x\in\mathbb{R}^3, \end{array}\right. $$ provided
that $V(r)$ has the following behavior at infinity: $$
  V(r)=V_0+\frac{a}{r^m}+O\left(\frac{1}{r^{m+\theta}}\right)
  \quad\mbox{ as } r\rightarrow\infty, $$ where $\frac12\le m<1$ and $a, V_0,
\theta$ are some positive constants. In particular, for any $s$ large we use a
reduction method to construct $s-$bump solutions lying on a circle of radius
$r\sim (s\log s)^{\frac{1}{1-m}}$.
