 Cauchy-kernel dressing for the KdV equation

Put $E_k=e^{i(kx+k^{3}t)}$, $I={\int }_L\phi (l)d\lambda (l)$ and $q=-I_x$. For the dressing operator $\mathcal{F}h=h+i{E}_k{\int }_{L}h(l)/(k+l)d\lambda (l)$, differentiating gives $\mathcal{F}(M\phi )=0$ and $\mathcal{F}(N\phi )=3ikM\phi$, where $M={\partial }_x^2-ik{\partial }_x+q$ and $N={\partial }_t+{\partial }_x^3+3q{\partial }_x$. Uniqueness of the linear integral equation therefore supplies a Lax pair. Its commutator identity is $[N,M]=-3q_{x}M+(q_t+q_{xxx}+6q{q}_x)$, which gives the KdV equation when the eigenfunction is not identically zero. Differentiation under the integral and the uniqueness assumption are essential hypotheses.