  In recent work (\cite{KW1},\cite{KW2}), Kostant and Wallach construct an
action of a simply connected Lie group $A\simeq \mathbb{C}^{{n\choose 2}}$ on
$gl(n)$ using a completely integrable system derived from the Poisson analogue
of the Gelfand-Zeitlin subalgebra of the enveloping algebra. In \cite{KW1}, the
authors show that $A$-orbits of dimension ${n\choose 2}$ form Lagrangian
submanifolds of regular adjoint orbits in $gl(n)$. They describe the orbit
structure of $A$ on a certain Zariski open subset of regular semisimple
elements. In this paper, we describe all $A$-orbits of dimension ${n\choose 2}$
and thus all polarizations of regular adjoint orbits obtained using
Gelfand-Zeitlin theory.
