Ulrich bundles on the degree six Segre fourfold

We study the resolution of an Ulrich bundle of arbitrary rank on the Segre fourfold ${\text{PP}}^2\times {\text{PP}}^2$. We characterize the Ulrich bundles $\text{Vv}$ of arbitrary rank on ${\text{PP}}^2\times {\text{PP}}^2$ with $h^1(\text{Vv}\otimes \mathrm{\Omega }⊠\mathrm{\Omega })=0$ or with $h^2(\text{Vv}\otimes \mathrm{\Omega }(-1)⊠\mathrm{\Omega }(-1))=0$ or obtained as pullback from ${\text{PP}}^2$ and we construct more complicated examples.