Another method for computing the densities of integrals of motion for the Korteweg–de Vries equation

In the first section of this article a new method for computing the densities of integrals of motion for the KdV equation is given. In the second section the variation with respect to $q$ of the functional $\int_0^\pi W(x,t,x;q)\,dx$ ($t$ is fixed) is computed, where $W(x,t,x;q)$ is the Riemann function of the problem
$$\begin{gather*} \frac{\partial^2u}{\partial x^2}-q(x)u=\frac{\partial^2u}{\partial t^2}\quad(-\infty<x<\infty), \\ u|_{t=0}f=(x),\quad\frac{\partial u}{\partial t}\Bigr|_{t=0}=0. \end{gather*}$$