  We prove the existence of infinitely many solutions $\lambda_1, \lambda_2 \in
\mathbb{R}$, $u,v \in H^1(\mathbb{R}^3)$, for the nonlinear Schr\"odinger
system \[ \begin{cases} -\Delta u - \lambda_1 u = \mu u^3+ \beta u v^2 &
\text{in $\mathbb{R}^3$} -\Delta v- \lambda_2 v = \mu v^3 +\beta u^2 v &
\text{in $\mathbb{R}^3$} u,v>0 & \text{in $\mathbb{R}^3$} \int_{\mathbb{R}^3}
u^2 = a^2 \quad \text{and} \quad \int_{\mathbb{R}^3} v^2 = a^2, \end{cases} \]
where $a,\mu>0$ and $\beta \le -\mu$ are prescribed. Our solutions satisfy
$u\ne v$ so they do not come from a scalar equation.
