  In this paper we show that there is a correspondence between some $K3$
surfaces with non-isometric transcendental lattices constructed as a twist of
the transcendental lattice of the Jacobian of a generic genus 2 curve.
Moreover, we show the existence of a correspondence between a general $K3$
surface with $\rho =17$ and a Kummer surface having transcendental lattices
$\QQ$-Hodge isomorphic.
