We present a new proof of the existence of symplectic realizations of a Poisson manifold $(M,\pi)$. The proof consists in the construction of a symplectic form on an open neighborhood of the zero-section of the cotangent bundle of $M$, defined by an explicit global formula.
Marius Crainic. Ioan Mǎrcuţ. "On the extistence of symplectic realizations." J. Symplectic Geom. 9 (4) 435 - 444, December 2011.