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 $\widehat{\mathfrak{g}}$ of $X_D$ and show that $X_D$ and $\widehat{\mathfrak{g}}$ are smooth and irreducible. We then use Poisson geometry to lift the Gelfand-Zeitlin vector fields on $X_D$ to Hamiltonian vector fields on $\widehat{\mathfrak{g}}$ and integrate these vector fields to an action of a connected, commutative algebraic group.