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In symplectic geometry a question of great importance is whether a(Lagrangian) submanifold is displaceable, that is, if it can be made disjointfrom itself by the means of a Hamiltonian isotopy. In these notes we analyzethe coadjoint orbits of SU(n) and their Lagrangian submanifolds that are fibersof the Gelfand-Tsetlin map. We use the coadjoint action to displace a largecollection of these fibers. Then we concentrate on the case n=3 and applyMcDuff's method of probes to show that "most" of the generic Gelfand-Tsetlinfibers are displaceable. "Most" means "all but one" in the non-monotone case,and means "all but a 1-parameter family" in the monotone case. In the case ofnon-monotone manifold of full flags we present explicitly an uniquenon-displaceable Lagrangian fiber (S^1)^3. This fiber was already proved to benon-displaceable in \cite{NNU}. Our contribution is in displacing other fibersand thus proving the uniqueness.

Displacing Lagrangians in the manifolds of full flags in C^3