Integrability and reduction of Hamiltonian actions on Dirac manifolds

For a Hamiltonian, proper and free action of a Lie group GG on a Dirac manifold (M,L)(M,L), with a regular moment map μ:M→g∗μ:M→g∗, the manifolds M/GM/G, μ−1(0)μ−1(0) and μ−1(0)/Gμ−1(0)/G all have natural induced Dirac structures. If (M,L)(M,L) is an integrable Dirac structure, we show that M/GM/G is always integrable, but μ−1(0)μ−1(0) and μ−1(0)/Gμ−1(0)/G may fail to be integrable, and we describe the obstructions to their integrability.