Summary: Associated to a constrained system with closed constraint algebra there are two Poisson manifolds \(P\) and \(Q\) forming a symplectic dual pair with respect to the original, unconstrained phase space: \(P\) is the image of the constraint map (equipped with the algebra of constraints) and \(Q\) the Poisson quotient with respect to the orbits generated by the constraints (the orbit space is assumed to be a manifold). We provide sufficient conditions so that the reduced phase space of the constrained system may be identified with a symplectic leaf of \(Q\). By these methods, a second class constrained system with closed algebra is reformulated as an abelian first class system in an extended phase space. While any Poisson manifold (\(P,\Phi\)) has a symplectic realization [M. Karasev, Math. USSR, Izv. 28, 497–527 (1987); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 50, No. 3, 508–538 (1986; Zbl 0624.58007) and A. Weinstein, Bull. Am. Math. Soc., New Ser. 16, 101–104 (1987; Zbl 0618.58020)], it does not always permit a leafwise symplectic embedding into a symplectic manifold (\(M,\omega\)). For regular \(P\), it is seen that such an embedding exists, iff the characteristic form-class of \(\Phi\), a certain element of the third relative cohomology of \(P\), vanishes. A tubular neighborhood of the constraint surface of a general second class constrained system equipped with the Dirac bracket provides a physical example for such an embedding into the original symplectic manifold. In contrast, a leafwise symplectic embedding of e.g. (the maximal regular part of) a Poisson Lie manifold associated to a compact, semisimple Lie algebra does not exist.