In this paper, we study the following nonlinear elliptic systems:
\begin{equation*} -\Delta u_1+V_1(x)u_1=\partial_{u_1}F(x,u)\quad x\in \mathbb{R}^N, \end{equation*}
\begin{equation*} -\Delta u_2+V_2(x)u_2=\partial_{u_2}F(x,u)\quad x\in \mathbb{R}^N, \end{equation*}
where $u=(u_1,u_2):\mathbb{R}^N\rightarrow \mathbb{R}^2$, $F$ and $V_i$ are periodic in $x_1,\cdots,x_N$ and $0\notin \sigma(-\Delta+V_i)$ for $i=1,2$, where $\sigma(-\Delta +V_i)$ stands for the spectrum of the Schr"odinger operator $-\Delta +V_i$.