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In this paper, we construct the additional flows of the noncommutativeKadomtsev-Petviashvili(KP) hierarchy and the additional symmetry flowsconstitute an infinite dimensional Lie algebra $W_{1+\infty}$. In addition, thegenerating function of the additional symmetries can also be proved to have anice form in terms of wave functions and this generating symmetry is used toconstruct the noncommutative KP hierarchy with self-consistent sources and theconstrained noncommutative KP hierarchy. The above results will be furthergeneralized to the noncommutative Gelfand-Dickey hierarchies which containsmany interesting noncommutative integrable systems such as the noncommutativeKdV hierarchy and noncommutative Boussinesq hierarchy. Meanwhile, we constructtwo new noncommutative systems including odd noncommutative C typeGelfand-Dickey and even noncommutative C type Gelfand-Dickey hierarchies. Alsousing the symmetry, we can construct a new noncommutative Gelfand-Dickeyhierarchy with self-consistent sources. Basing on the natural differential Laxoperator of the noncommutative Gelfand-Dickey hierarchy, the string equationsof the noncommutative Gelfand-Dickey hierarchy are also derived.

Symmetries and reductions on the noncommutative Kadomtsev-Petviashvili and Gelfand-Dickey hierarchies