Special Ulrich bundles on regular surfaces with non–negative Kodaira dimension

Let S be a regular surface endowed with a very ample line bundle OS(hS). Taking inspiration from a very recent result by D. Faenzi on K3 surfaces, we prove that if OS(hS) 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 PN. 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.