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

We consider the system of coupled elliptic equations $$\{\begin{array}{l}-\mathrm{\Delta }u-{\lambda }_{1}u={\mu }_1{u}^3+\beta u{v}^2\\ -\mathrm{\Delta }v-{\lambda }_{2}v={\mu }_2{v}^3+\beta u^{2}v\end{array}\text{in }{\mathbb{R}}^3,$$ and study the existence of positive solutions satisfying the additional condition $${\int }_{{\mathbb{R}}^3}{u}^2=a_1^2\text{and}{\int }_{{\mathbb{R}}^3}{v}^2=a_2^2\mathrm{.}$$ 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 is discussed, as well as the orbital stability of the corresponding standing waves for the related Schr"odinger systems.