Geometric reduction of Hamiltonian systems

Given a foliation S of a manifold M, a distribution Z in M transversal to S and a Poisson bivector Π on M, we present a geometric method of reducing this operator on the foliation S along the distribution Z. It encompasses the classical ideas of Dirac (Dirac reduction) and more modern theory of J. Marsden and T. Ratiu, but our method leads to formulae that allow for an explicit calculation of the reduced Poisson bracket. Moreover, we analyse the reduction of Hamiltonian systems corresponding to the bivector Π.