  We use momentum mappings on generalized flag manifolds and their momentum
polytopes to study the geometry of the level sets of the 1-chop integrals of
the full Kostant-Toda lattice in certain isospectral submanifolds of the phase
space. We derive expressions for these integrals in terms of Pl\"ucker
coordinates on the flag manifold in the case that all eigenvalues are zero and
compare the geometry of the base locus of their level set varieties with the
corresponding geometry for distinct eigenvalues. Finally, we illustrate and
extend our results in the context of the full sl(3,C) and sl(4,C) Kostant-Toda
lattices.
