Ulrich bundles on non-special surfaces with $p_g=0$ and $q=1$

Let $S$ be a surface with $p_g(S)=0$, $q(S)=1$ and endowed with a very ample line bundle ${\mathcal{O}}_S(h)$ such that $h^1(S,{\mathcal{O}}_S(h))=0$. We show that such an $S$ supports families of dimension $p$ of pairwise non-isomorphic, indecomposable, Ulrich bundles for arbitrary large $p$. Moreover, we show that $S$ supports stable Ulrich bundles of rank $2$ if the genus of the general element in $\mathrm{\mid }h\mathrm{\mid }$ is at least $2$.