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A surface $\Sigma$ endowed with a Poisson tensor $\pi$ is known to admit acanonical integration $\mathcal{G}(\pi)$, which is a 4-dimensional manifoldwith a (symplectic) groupoid structure. In this short note we show that when$\pi$ is not an area form on the 2-sphere, then $\mathcal{G}(\pi)$ isdiffeomorphic to the cotangent bundle $T^*\Sigma$, this extending results in\cite{Ma09} and \cite{BCST12}.

The diffeomorphism type of canonical integrations of Poisson tensors on surfaces