Explicit correspondences of a K3 surface with itself

Let $X$ be a K3-surface with a polarization $H$ of degree $H^2=2rs$, $r,s\ge1$. We consider the moduli space $Y$ of sheaves over $X$ with a primitive isotropic Mukai vector $(r,H,s)$. This space is again a K3-surface. In earlier papers, we gave necessary and sufficient conditions (in terms of the Picard lattice $N(X)$) for $Y$ and $X$ to be isomorphic. Here we show that these conditions imply the existence of an isomorphism between $Y$ and $X$ which is a composite of certain universal geometric isomorphisms between moduli of sheaves over $X$ and Tyurin's geometric isomorphism between moduli of sheaves over $X$ and $X$ itself. It follows that a general K3-surface $X$ with $\rho(X)=\operatorname{rk}N(X)\le2$ is isomorphic to $Y$ if and only if there is an isomorphism $Y\cong X$ which is a composite of universal isomorphisms and Tyurin's isomorphism.