  We prove that the surface $S(X)$ of bitangent lines of a general smooth
quartic surface $X$ in $\mP^3$ has unobstructed deformations of dimension
$20=h^1(S(X), T_{S(X)})$. In addition, we show that the space of infinitesimal
embedded deformations of $X$ injects into the one of $S(X)$. Finally we prove
that there is a natural birational map from the 20--dimensional moduli space of
(polarised) double coverings of EPW--sextics to the moduli space of regular
surfaces $S$ with $p_g=45$ and $K_S^2=360$ polarised with a very ample line
bundle $H$ such that $H^2=40$, $h^0(S, H)=6$: the map sends a double covering
of a EPW--sextic in $\mP^5$ to the surface of double points of the EPW--sextic.
