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, C). We use decomposition classes to stratify the strongly regular set by subvarieties XD. We construct an étale cover ĝD of XD and show that XD and ĝD are smooth and irreducible. We then use Poisson geometry to lift the Gelfand-Zeitlin vector fields on XD to Hamiltonian vector fields on ĝD and integrate these vector fields to an action of a connected, commutative algebraic group.