Special Ulrich bundles on non--special surfaces with $p_g=q=0$

Let $S$ be a surface with $p_g(S)=q(S)=0$ and endowed with a very ample line bundle $mathcal O_S(h)$ such that $h^1ig(S,mathcal O_S(h)ig)=0$. We show that $S$ supports special (often stable) Ulrich bundles of rank $2$, extending a recent result by A. Beauville. Moreover, we show that such an $S$ supports families of dimension $p$ of pairwise non--isomorphic, indecomposable, Ulrich bundles for arbitrary large $p$ except for very few cases. We also show that the same is true for each linearly normal non--special surface with $p_g(S)=q(S)=0$ in $p4$ of degree at least $4$, Enriques surface and anticanonical rational surface.