I am interested in Hamiltonian systems defined on manifolds which are the cotangent bundle $T^*M$ of a homogeneous manifold $M$.
Thus, I would like to find some reference in which it is studied when, starting from a homogeneous $G-$manifold $M$, we can generate a transitive group action even on $T^*M$ (I don't mean by cotangent-lifting it, since it is quite rare to lift it to a transitive action).
For the tangent bundle, I have studied (for example here Tangent bundle of homogeneous manifold) that if the action of $G$ on $M$ is transitive and, for any $g\in G$, the action of $g$ defines a submersion, then we can induce a transitive action on $TM$ by means of the Lie group $\bar{G}=G\ltimes \mathfrak{g}$. I started working on similar reasoning with $\bar{G}=G\ltimes \mathfrak{g}^*$, but for the moment it is not clear to me.
I believe that we should add the assumption that the action is free, in order to hope to define a similar action on $T^*M$.
Are there some known results for $T^*M$?
Another possible interesting perspective on this problem would be: when is the dual of a homogeneous manifold, homogeneous too? This would include more possible cases, but it would be really interesting for me.