Rank two nilpotent co-Higgs sheaves on complex surfaces

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