  We are concerned with the existence of solutions to the following nonlinear
Schr\"odinger system in $\mathbb{R}^3$: \begin{equation*} \left\{
\begin{aligned} -\Delta u_1 + (x_1^2+x_2^2)u_1&= \lambda_1 u_1 + \mu_1
|u_1|^{p_1 -2}u_1
  + \beta r_1|u_1|^{r_1-2}u_1|u_2|^{r_2}, \\ -\Delta u_2 + (x_1^2+x_2^2)u_2&=
\lambda_2 u_2 + \mu_2 |u_2|^{p_2 -2}u_2
  +\beta r_2 |u_1|^{r_1}|u_2|^{r_2 -2}u_2, \end{aligned} \right.
\end{equation*} under the constraint \begin{align*} \int_{\mathbb{R}^3}|u_1|^2
\, dx = a_1>0,\quad \int_{\mathbb{R}^3}|u_2|^2 \, dx = a_2>0, \end{align*}
where $\mu_1, \mu_2, \beta >0, 2 <p_1, p_2 < \frac{10}{3}$, $r_1, r_2>1, r_1 +
r_2 < \frac{10}{3}$. In the system, the parameters $\lambda_1, \lambda_2 \in
\R$ are unknown and appear as the associated Lagrange multipliers. Our
solutions are achieved as global minimizers of the underlying energy functional
subject to the constraint. Our purpose is to establish the compactness of any
minimizing sequence up to translations. As a by-product, we obtain the orbital
stability of the set of global minimizers.
