Question

I am having trouble trying to find a matrix $T$ so that with $X$, they form a Lax pair for the modified KdV equation $u_t - 6 u^2 u_x + u_{xxx} = 0$. Where $X$ is defined as:

$ X = \begin{pmatrix} \lambda & i u\\ - i u & - \lambda\\ \end{pmatrix}$

I have been told that $T_{22} = - T_{11}$, so let T:

$ T = \begin{pmatrix} a & b\\ c & - a\\ \end{pmatrix}$

Attempted solution.

So the using the compatibility condition $ \frac{\partial X}{\partial t} -\frac{\partial T}{\partial x} + [X,T] =0$ we get the following three equations.

$-a_x+i u (b+c) = 0$

$-2 i a u+2 b \lambda -b_x+i u_t = 0$

$-c_x-i \left(2 a u-2 i c \lambda +u_t\right) = 0$

However when I try to solve this I keep getting contradictions. Am I on the right path and if not can someone please help me understand how to find $T$.