  Constrained Hamiltonian systems fall into the realm of presymplectic
geometry. We show, however, that also Poisson geometry is of use in this
context.
  For the case that the constraints form a closed algebra, there are two
natural Poisson manifolds associated to the system, forming a symplectic dual
pair with respect to the original, unconstrained phase space. We provide
sufficient conditions so that the reduced phase space of the constrained system
may be identified with a symplectic leaf in one of those. In the second class
case the original constrained system may be reformulated equivalently as an
abelian first class system in an extended phase space by these methods.
  Inspired by the relation of the Dirac bracket of a general second class
constrained system to the original unconstrained phase space, we address the
question of whether a regular Poisson manifold permits a leafwise symplectic
embedding into a symplectic manifold. Necessary and sufficient for this is the
vanishing of the characteristic form-class of the Poisson tensor, a certain
element of the third relative cohomology.
