The Algebraic Geometry of the Kostant–Kirillov Form

Large classes of integrable Hamiltonian systems can be expressed as systems over coadjoint orbits in a loop algebra defined over a semi‐simple Lie algebra g. These systems can then be integrated via the classical, symplectic Liouville–Arnold method. On the other hand, the existence of spectral curves as constants of motion allows one to integrate these systems in terms of flows of line bundles on the curves. This note links the symplectic geometry of the coadjoint orbits with the algebraic geometry of these curves for arbitrary semi‐simple g, which then allows us to reconcile the two integration methods.