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The non-commutativ KdV equations are

$u_t +u_{xxx}-3(u u_x+ u_x u)=0$

They can be written by a pair of Lax :

$L(u)=-\partial^2 +u$

$A(u)=4 \partial^3-6u \partial -3 u_x$

$\frac {\partial L}{\partial t}=[L,A]$

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.

Yes, the noncommutative KdV equation in question is integrable, in particular in that it has infinitely many commuting symmetries and infinitely many conserved quantities which are in involution, as shown in the paper A new approach to the quantum KdV by Fuchssteiner and Chowdhury.

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