Towards noncommutative integrable systems

We present a powerful method to generate various equations which possess the Lax representations on noncommutative (1 + 1)- and (1 + 2)-dimensional spaces. The generated equations contain noncommutative integrable equations obtained by using the bicomplex method and by reductions of the noncommutative (anti-)self-dual Yang-Mills equation. This suggests that the noncommutative Lax equations would be integrable and be derived from reductions of the noncommutative (anti -)self-dual Yang-Mills equation, which implies the noncommutative version of Richard Ward conjecture. The integrability and the relation to string theories are also discussed. (C) 2003 Elsevier B.V. All rights reserved.