On irreducible symplectic 4-folds numerically equivalent to $(K3{)}^{[2]}$

We address the problem of classification of hyper-K"ahler fourfolds with $b_2=23$. In particular we prove some special cases of the Conjecture of O’Grady about hyper-K"ahler $4$-folds numerically equivalent to the Hilbert scheme of two points on a $K3$ surface.