  We construct nocommutative set-theoretical solutions to the Yang--Baxter
equation related to the KdV, the NLS and the derivative NLS equations. In
particular, we construct several Yang--Baxter maps of KdV type and we show that
one of them is completely integrable in the Liouville sense. Then, we construct
a noncommutative KdV type Yang--Baxter map which can be squeezed down to the
noncommutative discrete potential KdV equation. Moreover, we construct Darboux
transformations for the noncommutative derivative NLS equation. Finally, we
consider matrix refactorisation problems for noncommutative Darboux matrices
associated with the NLS and the derivative NLS equation and we construct
noncommutative maps. We prove that the latter are solutions to the Yang--Baxter
equation.
