  In the present paper we completely classify locally free sheaves of rank $2$
with vanishing intermediate cohomology modules on the image of the Segre
embedding $\mathbb{P}^2\times\mathbb{P}^2\subseteq\mathbb{P}^8$ and its general
hyperplane sections.
  Such a classification extends similar already known results regarding del
Pezzo varieties with Picard numbers $1$ and $3$ and dimension at least $3$.
