  In this paper, we give a complete study on the existence and non-existence of
normalized solutions for Schr\"{o}dinger system with quadratic and cubic
interactions. In the one dimension case, the energy functional is bounded from
below on the product of $L^2$-spheres, normalized ground states exist and are
obtained as global minimizers. When $N=2$, the energy functional is not always
bounded on the product of $L^2$-spheres. We give a classification of the
existence and nonexistence of global minimizers. Then under suitable conditions
on $b_1$ and $b_2$, we prove the existence of normalized solutions. When $N=3$,
the energy functional is always unbounded on the product of $L^2$-spheres. We
show that under suitable conditions on $b_1$ and $b_2$, at least two normalized
solutions exist, one is a ground state and the other is an excited state.
Furthermore, by refining the upper bound of the ground state energy, we provide
a precise mass collapse behavior of the ground state and a precise limit
behavior of the excited state as $\beta\rightarrow 0$. Finally, we deal with
the high dimensional cases $N\geq 4$. Several non-existence results are
obtained if $\beta<0$. When $N=4$, $\beta>0$, the system is a mass-energy
double critical problem, we obtain the existence of a normalized ground state
and its synchronized mass collapse behavior. Comparing with the well studied
homogeneous case $\beta=0$, our main results indicate that the quadratic
interaction term not only enriches the set of solutions to the above
Schr\"{o}dinger system but also leads to a stabilization of the related
evolution system.
