How can one tell whether a given (finite-dimensional) symplectic manifold is homogeneous (that is, admits a transitive group of symplectomorphisms)?

Note that it is of little help to observe that a homogeneous symplectic manifold must be a covering space of a coadjoint orbit of some Lie group, since this provides no effective test applicable to a given example. Note also that in the Riemannian case, there is a well-known local obstruction to the existence of a transitive group action by sutomorphisms: the covariant derivative of the curvature must vanish. But since all symplectic manifolds of dimension n are locally symplectomorphic, there can be no such obstruction in this case, and any analogous constraint must be simultaneously sensitive to the symplectic form and the topology of the manifold.