THE ORIGIN OF GAUGE SYMMETRIES IN INTEGRABLE SYSTEMS OF THE KDV TYPE

Generalized systems of integrable nonlinear differential equations of the KdV type are considered from the point of view of self-dual Yang-Mills theory in space-times with signature (2, 2). We present a systematic method for embedding the rth flows of the SL(N) KdV hierarchy with N greater-than-or-equal-to 2 and r < N in the dimensionally reduced self-dual system using SL(N) as gauge group. We also find that for r > N the corresponding equations can be described in a similar fashion, provided that (in general) the rank of the gauge group increases accordingly. Certain connections of this formalism with W(N) algebras are also discussed. Finally, we obtain a new class of nonlinear systems in two dimensions by introducing self-dual Ansatze associated with the W(N)(l) algebras of Bershadsky and Polyakov.