On a classical correspondence between K3 surfaces III

Let $X$ be a K3 surface, and $H$ its primitive polarization of the degree $H^2=8$. The moduli space of sheaves over $X$ with the isotropic Mukai vector $(2,H,2)$ is again a K3 surface, $Y$. In math.AG/0206158 we gave necessary and sufficient conditions in terms of Picard lattice of $X$ when $Y$ is isomorphic to $X$. The proof of sufficient condition in math.AG/0206158, when $Y$ is isomorphic to $X$, used Global Torelli Theorem for K3 surfaces, and it was not effective. Here we give an effective variant of these results: its sufficient part gives an explicit isomorphism between $Y$ and $X$. We hope that our similar results in math.AG/0304415, math.AG/0307355, math.AG/0309348 for arbitrary primitive isotropic Mukai vector on a K3 surface also can be made effective.