I was looking at the following interesting [question][1] 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?


  [1]: http://mathoverflow.net/questions/215067/equivariant-almost-complex-structures-on-the-full-flag-manifolds