  We analyze the geometric aspects of unitary evolution of general states for a
multilevel quantum system by exploiting the structure of coadjoint orbits in
the unitary group Lie algebra. Using the same method in the case of SU(3) we
study the effect of degeneracies on geometric phases for three-level systems.
This is shown to lead to a highly nontrivial generalization of the result for
two-level systems in which degeneracy results in a "monopole" structure in
parameter space. The rich structures that arise are related to the geometry of
adjoint orbits in SU(3). The limiting case of a two-level degeneracy in a
three-level system is shown to lead to the known monopole structure.
