  To each complex semisimple Lie algebra $\mathfrak{g}$ decorated with
appropriate data, one may associate two completely integrable systems. One is
the well-studied Kostant-Toda lattice, while the second is an integrable system
defined on the universal centralizer $\mathcal{Z}_{\mathfrak{g}}$ of
$\mathfrak{g}$. These systems are similar in that each exploits and closely
reflects the invariant theory of $\mathfrak{g}$, as developed by Chevalley,
Kostant, and others. One also has Kostant's description of level sets in the
Kostant-Toda lattice, which turns out to suggest deeper similarities between
the two integrable systems in question.
  We study relationships between the two aforementioned integrable systems,
partly to understand and contextualize the similarities mentioned above. Our
main result is a canonical open embedding of a flow-invariant open dense subset
of the Kostant-Toda lattice into $\mathcal{Z}_{\mathfrak{g}}$. Secondary
results include some qualitative features of the integrable system on
$\mathcal{Z}_{\mathfrak{g}}$.

Symplectic Geometry;Representation Theory