Correspondences of a K3 surface with itself via moduli of sheaves. I

Let $X$ be an algebraic K3 surface, $v=(r,H,s)$ a primitive isotropic Mukai vector on $X$ and $M_X(v)$ the moduli of sheaves over $X$ with $v$. Let $N(X)$ be Picard lattice of $X$. In math.AG/0309348 and math.AG/0606289, all divisors in moduli of $(X,H)$ (i. e. pairs $H\in N(X)$ with $\rk N(X)=2$) implying $M_X(v)\cong X$ were described. They give some Mukai's correspondences of $X$ with itself. Applying these results, we show that there exists $v$ and a codimension 2 submoduli in moduli of $(X,H)$ (i. e. a pair $H\in N(X)$ with $\rk N(X)=3$) implying $M_X(v)\cong X$, but this submoduli cannot be extended to a divisor in moduli with the same property. There are plenty of similar examples. We discuss the general problem of description of all similar submoduli and defined by them Mukai's correspondences of $X$ with itself and their compositions, trying to outline a possible general theory.