On Algebraic Integrability of Gelfand-Zeitlin fields

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.