KdV type equations from gauged WZW models and conformal-like gauge of W-gravity

We show that the Wess-Zumino-Witten models provide Toda and KdV type integrable equations when they are properly constrained. This allows us to connect soliton theories with the Chern-Simons type gauge theories at the classical level. Toda field equations appear as zero curvature conditions which again are the classical equations of motion of the constrained WZW actions. Bäcklund transformations appear naturally as gauge transformations and explicit transformation rules for Toda and KdV type equations are derived. We also show that the conformal-like gauge of W-gravity is related to the generalization of the Toda field theory.