I'm trying to understand what the conditions are for the Lax pairs for the zero-curvature representation:
$$ \partial_t U - \partial_x V + [U,V]=0 $$
where $U=U(x,t,\lambda)$ and $V=V(x,t,\lambda)$ are matrix-valued functions and $\lambda$ is a parameter.
The motivation behind this question is that the Lax pairs for the KdV equation:
$$ u_t + 6uu_x - u_{xxx} = 0 $$
is given by:
$$ U = \begin{pmatrix} 0 & 1 \\ \lambda + u & 0 \end{pmatrix} \text{ and } V = \begin{pmatrix} u_x & 4 \lambda - 2u \\ 4 \lambda^2 + 2 \lambda u + u_{xx} - 2u^2 & - u_x \end{pmatrix} $$
Now, it is not too difficult to verify that this indeed satisfies the zero-curvature representation, but I'm trying to figure out why we cannot use the Lax pairs:
$$ U = \begin{pmatrix} 0 & 0 \\ \lambda + u & 0 \end{pmatrix} \text{ and } V = \begin{pmatrix} 0 & 0 \\ \lambda + 3 u^2 - u_{xx} & 0 \end{pmatrix} $$
These matrices clearly satisfy the zero-curvature representation, but for some reason none of the notes I've been reading use them. What is the reason that they are not a valid Lax pair for the KdV equation?
I've also asked this question here (I hope that's ok): http://www.physicsoverflow.org/26475/integrability-conditions-of-lax-pairs