We know this important fact from A.A.Kirillov that :

Every homogeneous symplectic $G$-manifold is locally isomorphic to an orbit in the coadjoint representation of the group $G$ or a central extension of it.

But when can we precisely say that a homogeneous symplectic $G$-manifold is locally isomorphic to an orbit in the coadjoint representation of the group $G$

or

A homogeneous symplectic $G$-manifold is central extension of it?

PS:Definition of coadjoint orbit;

Let $G$ be a Lie Group and $\mathfrak{g}$ be its lie algebra,and also $\mathfrak{g^*}$ be the dual of Lie algebra, the coadjoint orbit is as follows

$\mathfrak{G}=\{Ad^*(g)F, g\in G\}$ where $F\in\mathfrak{g^*}$.