  Noncommutative Ward's conjecture is a noncommutative version of the original
Ward's conjecture which says that almost all integrable equations can be
obtained from anti-self-dual Yang-Mills equations by reduction. In this paper,
we prove that wide class of noncommutative integrable equations in both (2+1)-
and (1+1)-dimensions are actually reductions of noncommutative anti-self-dual
Yang-Mills equations with finite gauge groups, which include noncommutative
versions of Calogero-Bogoyavlenskii-Schiff eq., Zakharov system, Ward's chiral
and topological chiral models, (modified) Korteweg-de Vries, Non-Linear
Schroedinger, Boussinesq, N-wave, (affine) Toda, sine-Gordon, Liouville,
Tzitzeica, (Ward's) harmonic map eqs., and so on. This would guarantee
existence of twistor description of them and the corresponding physical
situations in N=2 string theory, and lead to fruitful applications to
noncommutative integrable systems and string theories. Some integrable aspects
of them are also discussed.
