We consider the quadratic Schrodinger system $$iu_t+Delta_{gamma_1}u+overline{u}v=0$$ $$2iv_t+Delta_{gamma_2}v-beta v+frac 12 u^2=0,$$ where $tinmathbf{R},,xin mathbf{R}^dtimes mathbf{R}$, in dimensions $1leq dleq 4$ and for $gamma_1,gamma_2>0$, the so-called elliptic-elliptic case. We show the formation of singularities and blow-up in the $L^2$-(super)critical case. Furthermore, we derive several stability results concerning the ground state solutions of this system.
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