  We describe a method for solving the Maurer-Cartan structure equation
associated with a Lie algebra that isolates the role of the Jacobi identity as
an obstruction to integration. We show that the method naturally adapts to two
other interesting situations: local symplectic realizations of Poisson
structures, in which case our method sheds light on the role of the Poisson
condition as an obstruction to realization; and the Maurer-Cartan structure
equation associated with a Lie algebroid, in which case we obtain an explicit
formula for a solution to the equation which generalizes the well known formula
in the case of Lie algebras.
