  Let $\mathcal{H}iggs_{ss}$ be the stack of semistable rank two Higgs bundles
on $\mathbf{P}^{1}$ with value in $O(n)$ where $n\geq 2$. In this paper we will
construct the Poincar\'e sheaf $\mathcal{P}$ on
$\mathcal{H}iggs_{ss}\times_{H}\mathcal{H}iggs_{ss}$ which is maximal
Cohen-Macaulay and flat over $\mathcal{H}iggs_{ss}$.
