Consider the following question
Consider the Poisson structure $$ \{F, G\}=\int_{\mathbb{R}} \frac{\delta F}{\delta u(x)} \frac{\partial}{\partial x} \frac{\delta G}{\delta u(x)} d x, $$ where $F, G$ are polynomial functionals of $u, u_x, u_{x x}, \ldots$ Assume that $u, u_x, u_{x x}, \ldots$ tend to zero as $|x| \rightarrow \infty$. (i) Show that $\{F, G\}=-\{G, F\}$. (ii) Write down Hamilton's equations for $u=u(x, t)$ corresponding to the following Hamiltonians: $$ H_0[u]=\int_{\mathbb{R}} \frac{1}{2} u^2 d x, \quad H[u]=\int_{\mathbb{R}}\left(\frac{1}{2} u_x{ }^2+u^3+u u_x\right) d x . $$ (iii) Calculate the Poisson bracket $\left\{H_0, H\right\}$, and hence or otherwise deduce that the following overdetermined system of partial differential equations for $u=u\left(x, t_0, t\right)$ is compatible: $$ \begin{gathered} u_{t_0}=u_x, \\ u_t=6 u u_x-u_{x x x} . \end{gathered} $$ [You may assume that the Jacobi identity holds for (1).]
I was able to do the first two parts of the question, where note that hamiltons equations with respect tothe hamiltonians turn out to be (2) and (3). I was also able to, by force show that $$ \{H_0,H\}=0. $$
However, I am not sure how to proceed. Per the question I assume that I need to use the Jacobi identity $$\{F,\{G, H\}\}+\{G,\{H, F\}\}+\{H,\{F, G\}\}=0.$$ This suggests we set two of the hamiltonians be the one in the question, but I do not have an idea as to what should be the third.
Moreover, I do not really understand what the question is asking of me. Am I asked to show that $$ u_t= u_{t_0}? $$
Question: What is being asked of me in the end of (iii) and how does one approach it?