  Let $(\mathcal{E}, \phi)$ be a rank two co-Higgs vector bundles on a K\"ahler
compact surface $X$ with $\phi\in H^0(X,End(\mathcal{E})\otimes T_X)$
nilpotent. If $(\mathcal{E}, \phi)$ is semi-stable, then one of the following
holds up to finite \' etale cover:
  $i)$ $X$ is uniruled.
  $ii)$ $X$ is a torus and $(\mathcal{E}, \phi)$ is strictly semi-stable.
  $iii)$ $X$ is a properly elliptic surface and $(\mathcal{E}, \phi)$ is
strictly semi-stable.
