On a classical correspondence between K3 surfaces
  Let X be a K3 surface which is intersection of three (a net P^2) of quadrics in P^5. The curve of degenerate quadrics has degree 6 and defines a double covering of P^2 K3 surface Y ramified in this curve. This is a classical example of a correspondence between K3 surfaces which is related with moduli of vector bundles on K3 studied by Mukai. When general (for fixed Picard lattices) X and Y are isomorphic? We give necessary and sufficient conditions in terms of Picard lattices of X and Y.   E.g. for Picard number 2, the Picard lattice of X and Y is defined by its determinant (-d) where d>0, d\equiv 1 \mod 8, and one of equations a^2-db^2=8 or a^2-db^2=-8 should have an integral solution (a,b). The set of these d is infinite: d\in {(a^2\mp 8)/b^2} where a and b are odd integers. This describes all possible divisorial conditions on 19- dimensional moduli of intersections of three quadrics in P^5 when Y\cong X. One of them when X has a line is classical, and corresponds to d=17.   Similar considerations can be applied for a realization of an isomorphism (T(X)\otimes Q, H^{2,0}(X)) \cong (T(Y)\otimes Q, H^{2,0}(Y)) of transcendental periods over Q of two K3 surfaces X and Y by a fixed sequence of types of Mukai vectors. 
Carlo Madonna and Viacheslav V. Nikulin