  Let $S$ be a regular surface endowed with a very ample line bundle $\mathcal
O_S(h_S)$. Taking inspiration from a very recent result by D. Faenzi on $K3$
surfaces, we prove that if $\mathcal O_S(h_S)$ satisfies a short list of
technical conditions, then such a polarized surface supports special Ulrich
bundles of rank $2$. As applications, we deal with general embeddings of
regular surfaces, pluricanonically embedded regular surfaces and some properly
elliptic surfaces of low degree in $\mathbb P^N$. Finally, we also discuss
about the size of the families of Ulrich bundles on $S$ and we inspect the
existence of special Ulrich bundles on surfaces of low degree.
