  We look for ground state solutions to the Sch\"odinger-type system
  \[
  \begin{cases}
  -\Delta u_j+\lambda_ju_j=\partial_jF(u)
  \int_{\mathbb{R}^N}u_j^2\,dx=a_j^2
  (\lambda_j,u_j)\in\mathbb{R}\times H^1(\mathbb{R}^N)
  \end{cases} j\in\{1,\dots,M\}
  \] with $N\ge1$ and $1\le M<2+4/N$, 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 generic assumptions on the nonlinearity $F$ which
correspond to the $L^2$-subcritical and $L^2$-critical cases, i.e., when the
energy is bounded from below for all or some values of $a$. Making use of a
recent idea, we minimize 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 $|u_j|_{L^2}=a_j$.
