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Description

We define a class of Poisson manifolds that is well-behaved from the point of view of singular foliations, understood as as submodules of vector fields rather than partitions into leaves: the class of almost regular Poisson manifolds. The latter admit a geometric characterization in terms of the symplectic leaves alone, and contain the class of log-symplectic manifolds. We study the holonomy groupoid integrating the singular foliation of an almost regular Poisson structure. We shot that it is a Poisson groupoid, integrating a naturally associated Lie bialgebroid. The Poisson structure on the holonomy groupoid is regular, and as such it provides a desingularization.