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