Self-Dual Yang-Mills and the Hamiltonian Structures of Integrable Systems

In recent years it has been shown that many, and possibly all, integrable systems can be obtained by dimensional reduction of self-dual Yang-Mills. I show how the integrable systems obtained this way naturally inherit bihamiltonian structure. I also present a simple, gauge-invariant formulation of the self-dual Yang-Mills hierarchy proposed by several authors, and I discuss the notion of gauge equivalence of integrable systems that arises from the gauge invariance of the self-duality equations (and their hierarchy); this notion of gauge equivalence may well be large enough to unify the many diverse existing notions.