The non-commutativ KdV equations are
ut+uxxx−3(uux+uxu)=0
They can be written by a pair of Lax :
L(u)=−∂2+u
A(u)=4∂3−6u∂−3ux
∂L∂t=[L,A]
Is it an integrable system? For example, u=a+bi+cj+dk has values in the Hamilton numbers R[i,j,k]; it is then equivalent to a set of four coupled real differential equations.