On the geometry of certain isospectral sets in the full Kostant-Toda lattice

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.