  In this short note we give a complete characterization of a certain class of
compact corank one Poisson manifolds, those equipped with a closed one-form
defining the symplectic foliation and a closed two-form extending the
symplectic form on each leaf. If such a manifold has a compact leaf, then all
the leaves are compact, and furthermore the manifold is a mapping torus of a
compact leaf. These manifolds and their regular Poisson structures admit an
extension as the critical hypersurface of a Poisson b-manifold as we consider
in a later paper.
