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We consider the system of coupled elliptic equations \[ \begin{cases} -\Deltau - \lambda_1 u = \mu_1 u^3+ \beta u v^2 \\ -\Delta v- \lambda_2 v = \mu_2 v^3+\beta u^2 v \end{cases} \text{in $\mathbb{R}^3$}, \] and study the existenceof positive solutions satisfying the additional condition \[\int_{\mathbb{R}^3} u^2 = a_1^2 \quad \text{and} \quad \int_{\mathbb{R}^3} v^2= a_2^2. \] Assuming that $a_1,a_2,\mu_1,\mu_2$ are positive fixed quantities,we prove existence results for different ranges of the coupling parameter$\beta>0$. The extension to systems with an arbitrary number of components isdiscussed, as well as the orbital stability of the corresponding standing wavesfor the related Schr\"odinger systems.

Normalized solutions for a system of coupled cubic Schr\"odinger equations on $\mathbb{R}^3$