  We use the Gelfand-Tsetlin pattern to construct an effective Hamiltonian,
completely integrable action of a torus T on an open dense subset of a
coadjoint orbit of the unitary group. We then identify a proper Hamiltonian
T-manifold centered around a point in the dual of the Lie algebra of T. A
theorem of Karshon and Tolman says that such a manifold is equivariantly
symplectomorphic to a particular subset of R^2D. This fact enables us to
construct symplectic embeddings of balls into certain coadjoint orbits of the
unitary group, and therefore obtain a lower bound for their Gromov width. Using
the identification of the dual of the Lie algebra of the unitary group with the
space of (n x n) Hermitian matrices, the main theorem states that for a
coadjoint orbit through \lambda=diag(\lambda_1, ..., \lambda_n) in the dual of
the Lie algebra of the unitary group, where at most one eigenvalue is repeated,
the lower bound for Gromov width is equal to the minimum of the differences
\lambda_i-\lambda_j, over all \lambda_i>\lambda_j. For a generic orbit (i.e.
with distinct \lambda_i's), with additional integrality conditions, this
minimum has been proved to be exactly the Gromov width of the orbit. For
nongeneric orbits this lower bound is new.
