Integrability and Reduction of Hamiltonian Actions on Dirac Manifolds

For a Hamiltonian, proper and free action of a Lie group $G$ on a Dirac manifold $(M,L)$, with a regular moment map $\mu:M\to \mathfrak{g}^*$, the manifolds $M/G$, $\mu^{-1}(0)$ and $\mu^{-1}(0)/G$ all have natural induced Dirac structures. If $(M,L)$ is an integrable Dirac structure, we show that $M/G$ is always integrable, but $\mu^{-1}(0)$ and $\mu^{-1}(0)/G$ may fail to be integrable, and we describe the obstructions to their integrability.