  We study the existence and stability of standing waves for a system of
nonlinear Schr\"odinger equations with quadratic interaction in dimensions
$d\leq 3$. We also study the characterization of finite time blow-up solutions
with minimal mass to the system under mass resonance condition in dimension
$d=4$. Finite time blow-up solutions with minimal mass are showed to be (up to
symmetries) pseudo-conformal transformations of a ground state standing wave.
