Stability of rank two Ulrich bundles on projective K3 surfaces

Let $F\subseteq \mathbb{P}^{N}$ be a $K3$ surface of degree $2a$, where $a\ge 2$. In this paper we deal with Ulrich bundles on $F$ of rank $2$. We deal with their stability and we construct $K3$ surfaces endowed with families of non-special Ulrich bundles of rank $2$ for each $a\ge 2$.