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The nonlinear Schr\"{o}dinger-Newton system \begin{equation*}  \begin{cases}  \Delta u- V(|x|)u + \Psi u=0, &~x\in\mathbb{R}^3,\\ \Delta \Psi+\frac12u^2=0, &~x\in\mathbb{R}^3, \end{cases} \end{equation*} is a nonlinear systemobtained by coupling the linear Schr\"{o}dinger equation of quantum mechanicswith the gravitation law of Newtonian mechanics. Wei and Yan in (Calc. Var.Partial Differential Equations 37 (2010),423--439) proved that theSchr\"{o}dinger equation has infinitely many positive solutions in$\mathbb{R}^N$ and these solutions have polygonal symmetry in the $(y_{1},y_{2})$ plane and they are radially symmetric in the other variables. Duan etal. in (arXiv:2006.16125v1) extended the results got by Wei and Yan and thesesolutions have polygonal symmetry in the $(y_{1}, y_{2})$ plane and they areeven in $y_{2}$with one more more parameter in the expression of thesolutions.Hu et al. Under the appropriate assumption on the potential functionV, Hu et al. in (arXiv: 2106.04288v1) constructed infinitely many non-radialpositive solutions for the Schr\"{o}dinger-Newton system and these positivesolutions have polygonal symmetry in the $(y_{1}, y_{2})$ plane and they areeven in $y_{2}$ and $y_{3}$. Assuming that $V(r)$ has the following character\begin{equation*}V(r)=V_{1}+\frac{b}{r^q}+O\Big(\frac{1}{r^{q+\sigma}}\Big),~\mbox{ as }r\rightarrow\infty, \end{equation*} Where $\frac12\leq q<1$ and $b, V_{1},\sigma$ are some positive constants, $V(y)\geq V_1>0$, we construct infinitelymany non-radial positive solutions which have polygonal symmetry in the$(y_{1}, y_{2})$ plane and are even in $y_{2}$ for the Schr\"{o}dinger-Newtonsystem by the Lyapunov-Schmidt reduction method. We extend the results got byDuan et al. in (arXiv:2006.16125v1) to the nonlinear Schr\"{o}dinger-Newtonsystem.

New type of solutions for the nonlinear Schr\"odinger-Newton system