  We prove that the Hilbert square $S^{[2]}$ of a very general primitively
polarized K3 surface S of degree $d(n) = 2(4n^2 + 8n + 5)$, $n \geq 1$ is
birational to a double Eisenbud-Popescu-Walter sextic. Our result implies a
positive answers, in the case when $r$ is even, to a conjecture of O'Grady: On
the Hilbert square of a very general K3 surface of genus $r^2 + 2$, $r \geq 1$
there is an antisymplectic involution. We explicitly give this involution on
$S^{[2]}$ in term of the corresponding EPW polarization on it.
