  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.
