THE TWISTOR-THEORY OF EQUATIONS OF KDV TYPE .1.

This article is the first of two concerned with the development of the theory of equations of KdV type from the point of view of twistor the ory and the self-dual Yang-Mills equations. A hierarchy on the self-du al Yang-Mills equations is introduced and it is shown that a certain r eduction of this hierarchy is equivalent to the n-generalized KdV-hier archy. It also emerges that each flow of the n-KdV hierarchy is a redu ction of the self-dual Yang-Mills equations with gauge group SL(n). It is further shown that solutions of the self-dual Yang-Mills hierarchy and their reductions arise via a generalized Ward transform from holo morphic vector bundles over a twistor space. Explicit examples of such bundles are given and the Ward transform is implemented to yield a la rge class of explicit solutions of the n-KdV equations. It is also sho wn that the construction of Segal and Wilson of solutions of the n-KdV equations from loop groups is contained in our approach as an ansatz for the construction of a class of holomorphic bundles on twistor spac e. A summary of the results of the second part of this work appears in the Introduction.