I was looking at the following interesting [question](https://mathoverflow.net/questions/215067/equivariant-almost-complex-structures-on-the-full-flag-manifolds) about the number of equivariant almost complex structures on the full flag manifold of $SU(3)$, and I began to wonder how many equivariant symplectic structures there are on the same space, i.e. $M := SU(3)/T^2$. Now as I understand it, there is a unique equivariant K\"ahler metric for $M$, which will of course give us one equivariant symplectic structure, and since symplectic structures which do not arise from K\"ahler metrics are hard to come by, I would conjecture that this is the only equivariant symplectic structure for this space. Is this true?



Unique Equivariant Symplectic Structure for the Full Flag Manifold of $SU(3)$?
If $G$ acts on $M$ (both compact and finite-dimensional) preserving the symplectic form, and $M$ is simply-connected, the action is Hamiltonian. Then $M$ maps symplectomorphically to a coadjoint orbit of $G$. This gives a $rank(G)$-dimensional family of symplectic forms, not a unique one.


You should consider the $SU(2)$ case first, where $\mathbb{CP}^1$ doesn't have a unique area form (or area).


