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We look for ground state solutions to the Schr\"odinger-type system \[\begin{cases} -\Delta u_j + \lambda_j u_j = \partial_jF(u)\\ \int_{\rn} u_j^2\, dx = a_j^2\\ (\lambda_j,u_j) \in \mathbb{R} \times H^1(\mathbb{R}^N)\end{cases} j \in \{1,\dots,M\} \] with $N,M\ge1$, where $a=(a_1,\dots,a_M) \in]0,\infty[^M$ is prescribed and $(\lambda,u) =(\lambda_1,\dots,\lambda_M,u_1,\dots u_M)$ is the unknown. We provide genericassumptions about the nonlinearity $F$ which correspond to the$L^2$-subcritical and $L^2$-critical cases, i.e., when the energy is boundedfrom below for all or some values of $a$. Making use of a recent idea, weminimize the energy over the constraint $\Set{\left|u_j\right|_{L^2}\le a_j\text{ for all } j}$ and then provide further assumptions that ensure$|u_j|_{L^2}=a_j$.

Normalized ground states to a cooperative system of Schr\"odinger equations with generic $L^2$-subcritical or $L^2$-critical nonlinearity