On a classical correspondence between K3 surfaces II

Let X be a K3 surface and H a primitive polarization of degree H^2=2a^2, a>1. The moduli space of sheaves over X with the isotropic Mukai vector (a,H,a) is again a K3 surface Y which is endowed by a natural nef element h with h^2=2. We give necessary and sufficient conditions in terms of Picard lattices N(X) and N(Y) when Y\cong X, generalising our results math.AG/0206158 for a=2. E.g. we show that Y\cong X if for one of \alpha =\pm 1,\pm 2 which is coprime to a there exists h_1\in N(X) such that h_1^2= 2\alpha a, H\cdot h_1\equiv 0\mod \alpha a, and the primitive sublattice [H,h_1]_{pr} \subset N(X) contains x such that $x\cdot H=1$. We find all divisorial conditions on moduli of (X,H) (i.e for Picard number 2) which imply Y\cong X and H\cdot N(X)=Z. Some of these conditions were found in different form by A.N. Tyurin in 1987.