manifold, the image is a regular distribution; it is easy to check that it is involutive, therefore, by Frobenius theorem, admits a partition into leaves. Moreover, the Poisson bivector restricts nicely to each leaf, which become therefore symplectic manifolds. The non-regular case For a non-regular Poisson manifold the situation is more complicated, since the distribution is singular, i.e. the vector subspaces have different dimensions. An integral submanifold for is a path-connected submanifold satisfying for all . Integral submanifolds of are automatically regularly immersed manifolds, and maximal integral submanifolds of are called the leaves of . Moreover, each leaf carries a natural symplectic form determined by the condition for all and . Correspondingly, one speaks of the symplectic leaves of . Moreover, both the space of regular points and its complement are saturated by symplectic leaves, so symplectic leaves may be either regular or singular. Weinstein splitting theorem To show the existence of symplectic leaves also in the non-regular case, one can use Weinstein splitting theorem (or Darboux-Weinstein theorem). It states that any Poisson manifold splits locally around a point as the product of a symplectic manifold and a transverse Poisson submanifold vanishing at . More precisely, if , there are local coordinates such that the Poisson bivector splits as the sumwhere . Notice that, when the rank of is maximal (e.g. the Poisson structure is nondegenerate), one recovers the classical Darboux theorem for symplectic structures. Examples Trivial Poisson structures Every manifold carries the trivial Poisson structure , equivalently described by the bivector . Every point of is therefore a zero-dimensional symplectic leaf. Nondegenerate Poisson structures A bivector field is called nondegenerate if is a vector bundle isomorphism. Nondegenerate Poisson bivector fields are actually the same thing as symplectic manifolds . Indeed, there is a bijective correspondence between nondegenerate bivector fields and nondegenerate 2-forms , given bywhere is encoded by . Furthermore, is Poisson precisely if and only if is closed; in such case, the bracket becomes the canonical Poisson bracket from Hamiltonian mechanics:Non-degenerate Poisson structures have only one symplectic leaf, namely itself, and their Poisson algebra become a Poisson ring. Linear Poisson structures A Poisson structure on a vector space is called linear when the bracket of two linear functions is still linear. The class of vector spaces with linear Poisson structures coincides actually with that of (dual of) Lie algebras. Indeed, the dual of any finite-dimensional Lie algebra carries