Generalized Toda theories and -algebras associated with integral gradings

A general class of conformal Toda theories associated with integral gradings of Lie algebras is investigated. These generalized Toda theories are obtained by reducing the Wess--Zumino--Novikov--Witten (WZNW) theory by first--class constraints, and thus they inherite extended conformal symmetry algebras, generalized W--algebras, and current dependent Kac--Moody (KM) symmetries from the WZNW theory, which are analysed in detail in a non--degenerate case. We recover an $sl(2)$ structure underlying the generalized W--algebras, which allows for identifying the primary fields, and give a simple algorithm for implementing the W--symmetries by current dependent KM transformations, which can be used to compute the action of the W--algebra on any quantity. We establish how the Lax pair of Toda theory arises in the WZNW framework, and show that a recent result of Mansfield and Spence, which interprets the W--symmetry of the Toda theory by means of non--Abelian form preserving gauge transformations of the Lax pair, arises immediately as a consequence of the KM interpretation