thank you @Dilaton, i'm going to reply to that answer. here i just remark that, while i agree with Arnold and found his reference very useful, it doesn't quite solve my problem, yet.

@Igor Yes I was not very specific in the motivation, I wanted just to to give an idea. What I had in mind was more the non-relativistic hydrodynamic Poisson bracket $\{\overline{f},\overline{g}\} = \int \frac{\delta \overline{f}}{\delta u } K( \frac{\delta \overline{g}}{\delta u} )dx$, with $K = \sum K_i \partial_x^i$ a differential operator with coefficients $K_i(u,u_x,u_{xx},\ldots)$ that are diff. polynomials and $\overline{f}=\int f(u,u_x,u_{xx},\ldots)dx$, with $x\in S^1$. People consider quantization of such systems (KdV, etc) all the time and I have been wandering about its unicity.

In a nutshell the monodromy matrix in the rapidly decreasing case (what DKN call the scattering theory) is the transition matrix between two basis of solutions to the spectral problem $L \psi = \lambda \psi$: the basis of solutions that are asymptotically periodic at $-\infty$ and $+\infty$, respectively. Such monodromy plays a similar role to what you defined for the periodic case, in particular its trace is a constant of motion. Anyway the rapidly decreasing and periodic cases are both treated in the reference I gave you. It's a really good read.

Yes, you are right that the rapidly decreasing case and the periodic case deserve a somwhat separate treatment. I quote from "Integrable Systems. I" by B.A. Dubrovin, I.M. Krichever, S.P. Novikov, which you can find in "Dynamical Systems IV" and which I recommend as one of the best references: "In the periodic case, the spectral theory is completely different and bears no resemblance to the scattering theory".

in order for the conserved quantities to be expressed as local functionals $F[u] = \int f(u,u_x,u_{xx},\ldots) dx$ you need to have boundary condition which allow to make sense of the integral. Periodicity is not always required, you can ask that $u(x) \to 0$ fast enough as $x\to \pm \infty$. Or you can define $u(x)$ on a circle ($x\in S^1$) so that integration makes sense and integrating by parts has no boundary term.