Let \(X\) be a Fano threefold, and let \(S\subset X\) be a \(K3\) surface. Let \(\mathcal{M}_X\) be an irreducible component of a moduli space of simple vector bundles on \(X\) such that their restriction to \(S\) belongs to a moduli space \(M_S\). In a 1990 preprint [The moduli spaces of vector bundles on threefolds, surfaces and curves I. in: Vector bundles. Göttingen: Universitätsverlag Göttingen. 176–213 (2008)], A. Tyurin remarked that if \(\mathrm{H}^2(X,\mathrm{End}(E)) = 0\) for all \(E\in \mathcal{M}_X\), then the restriction map is a local isomorphism to a Lagrangian subvariety of \(\mathcal{M}_S\) (recall that \(\mathcal{M}_S\) has a natural holomorphic symplectic structure).
In this paper the author provides many examples where this result can be applied, focussing on the cases where \(X\) is a prime Fano threefold of index 1 or 2, and \(S\) is a smooth surface in the anticanonical system of \(X\). Then by using the Serre construction and working with rank 2 vector bundles, the author describes how to obtain Lagrangian subvarieties of \(\mathcal{M}_S\).
More precisely, the following cases (under suitable further conditions) are considered:

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\(X\subset \mathbb{P}^4\) a cubic threefold, \(M_X\) is the Fano surface of lines contained in \(X\) and \(M_S=S^{[2]}\);

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\(X\subset \mathbb{P}^5\) the intersection of two quadrics;

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\(X\subset \mathbb{P}^6\) a Fano threefold of index 2 and degree 5, \(M_X\) and \(M_S\) consist of one reduced point;

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\(X_d\subset \mathbb{P}^d\) a Fano threefold of index 2 and degree \(d = 3, 4, 5\), \(M_S\) is the O’Grady hyperkähler manifold \(OG_{10}\);

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\(X\subset \mathbb{P}^6\) intersection of three quadrics: in this case the image of the restriction map from \(M_X\) to \(M_S\) is singular;

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\(X_g\subset \mathbb{P}^{g+1}\) a Fano threefold of index 1 and genus \(g\): then \(M_S=OG_{10}\).