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In this paper, we study the normalized solutions of the Schr\"{o}dinger system with trapping potentials \begin{equation}\label{eq:diricichlet} \begin{cases} -\Delta u_1+V_1(x)u_1-\lambda_1 u_1=\mu_1 u_1^3+\beta u_1u_2^{2}+\kappa u_2~\hbox{in}~ \mathbb{R}^3,\\ -\Delta u_2+V_2(x)u_2-\lambda_2 u_2=\mu_2 u_2^3+\beta u_1^2u_2+\kappa u_1~\hbox{in}~ \mathbb{R}^3, u_1\in H^1(\mathbb{R}^3), u_2\in H^1(\mathbb{R}^3),\nonumber \end{cases} \end{equation} under the constraint \begin{equation} \int_{\mathbb{R}^3} u_1^2=a_1^2,~\int_{\mathbb{R}^3} u_2^2=a_2^2\nonumber, \end{equation} where $\mu_1,\mu_2,a_1,a_2,\beta>0$, $\kappa\in\mathbb{R}$, $V_1(x)$ and $V_2(x)$ are trapping potentials, and $\lambda_1,\lambda_2$ are lagrangian multipliers, this is a typical $L^2$-supercritical case in $\mathbb{R}^3$. We obtain the existence of solutions to this system by minimax theory on the manifold for $\kappa=0$ and $\kappa\neq 0$ respectively.

In this paper, we study important Schr\"{o}dinger systems with linear and nonlinear couplings \begin{equation}\label{eq:diricichlet} \begin{cases} -\Delta u_1-\lambda_1 u_1=\mu_1 |u_1|^{p_1-2}u_1+r_1\beta |u_1|^{r_1-2}u_1|u_2|^{r_2}+\kappa (x)u_2~\hbox{in}~\mathbb{R}^N,\\ -\Delta u_2-\lambda_2 u_2=\mu_2 |u_2|^{p_2-2}u_2+r_2\beta |u_1|^{r_1}|u_2|^{r_2-2}u_2+\kappa (x)u_1~ \hbox{in}~\mathbb{R}^N,\\ u_1\in H^1(\mathbb{R}^N), u_2\in H^1(\mathbb{R}^N),\nonumber \end{cases} \end{equation} with the condition $$\int_{\mathbb{R}^N} u_1^2=a_1^2, \int_{\mathbb{R}^N} u_2^2=a_2^2,$$ where $N\geq 2$, $\mu_1,\mu_2,a_1,a_2>0$, $\beta\in\mathbb{R}$, $2