  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.
