Integrable symplectic structures on compact complex manifolds

The following question is studied. Suppose one is given a $2n$-dimensional compact complex manifold with holomorphic symplectic 2-form. Are there obstructions to the existence of $n$ independent meromorphic first integrals in involution, and if so, what are they like? The answer to this question is given for K3 surfaces, Beauville manifolds, and complex tori; in these cases there are obstructions of an analytic character. Whether there are any topological obstructions is an unsolved problem.
Bibliography: 18 titles.