  We generalize a result of Kostant and Wallach concerning the algebraic
integrability of the Gelfand-Zeitlin vector fields to the full set of strongly
regular elements in $gl(n,\mathbb{C})$. We use decomposition classes to
stratify the strongly regular set by subvarieties $X_{D}$. We construct an
\'{e}tale cover $\hat{\mathfrak{g}}$ of $X_{D}$ and show that $X_{D}$ and
$\hat{\mathfrak{g}}$ are smooth and irreducible. We then use Poisson geometry
to lift the Gelfand-Zeitlin vector fields on $X_{D}$ to Hamiltonian vector
fields on $\hat{\mathfrak{g}}$ and integrate these vector fields to an action
of a connected, commutative algebraic group.
