Vector bundles on Fano threefolds and K3 surfaces

Let $X$ be a Fano threefold, and let $S \subset X$ be a smooth anticanonical surface (hence a K3). Any moduli space $\mathcal{M}_S$ of simple vector bundles on $S$ carries a holomorphic symplectic structure. Following an idea of Tyurin, I will show that in some cases, those vector bundles which come from $X$ form a Lagrangian subvariety of $\mathcal{M}_S$. Most of the talk will be devoted to concrete examples of this situation.