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We give many examples in which there exist infinitely many divisorialconditions on the moduli space of polarized K3 surfaces $(S,H)$ of degree$H^2=2g-2$, $g \geq 3$, and Picard number $rk N(S)=\rho(S)=2$ such that for ageneral K3 surface $S$ satisfying these conditions the moduli space of sheaves$M_S(r,H,s)$ is birationally equivalent to the Hilbert scheme $S[g-rs]$ ofzero-dimensional subschemes of $S$ of lenght equal to $g-rs$. This resultgeneralizes the main result of \cite{Nik1} when $g=rs+1$ and of \cite{Monat}when $r=s=2$, $g \geq 5$.

On some moduli spaces of bundles on K3 surfaces, II