  In this paper we establish the existence of canonical coordinates for generic
co-adjoint orbits on triangular groups. These orbits correspond to a set of
full Plancherel measure on the associated dual groups. This generalizes a
well-known coordinatization of co-adjoint orbits of a minimal (non-generic)
type originally discovered by Flaschka. The latter had strong connections to
the classical Toda lattice and its associated Poisson geometry. Our results
develop connections with the Full Kostant-Toda lattice and its Poisson
geometry. This leads to novel insights relating the details of Plancherel
theorems for Borel Lie groups to the invariant theory for Borels and their
subgroups. We also discuss some implications for the quantum integrability of
the Full Kostant Toda lattice.
