Once I know a complete integrable system $(f_1=H,\ldots,f_m):M^{2m}\to\mathbb{R}$, $H$ being the Hamiltonian of the system. What is the importance to know about a second Hamiltonian representation for the system? I would like to know what kind of information\work (research directions) can be done do with a bi-Hamiltonian structure of a system.
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1$\begingroup$ The main use of a bihamiltonian structure is the recursion relation that allows you to start with one of the $f_j$ and construct the others, but if you already have the $f_j$, then knowing that the system is bi-hamiltonian might not be so useful; although I look forward to being surprised by other answers! $\endgroup$– José Figueroa-O'FarrillApr 11, 2016 at 22:38
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1$\begingroup$ Thanks for the comment Jose!! I would like to know if there are other answers to this question too. $\endgroup$– user90218Apr 12, 2016 at 13:58
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While José's comment pretty much answers this, let me add that, sometimes, it can be hard or impossible to have the system of commuting symmetries $f_j$ under control, and the bihamiltonian recursion helps encoding and sorting them.
An example is when your phase space is actually infinite dimensional (here the case of Korteweg-De Vries equation is a canonical reference). It is not always possible to have some explicit generating series for the infinite number of commuting symmetries $f_j$ in such cases, but knowing the explicit form of the two compatible Poisson structures effectively encodes the information to reconstruct them all. This is maybe why bihamiltonian structures have gained much more preeminence in the integrable evolutionary PDEs/infinite dimensional integrable systems world than the more classical, finite dimensional Arnold-Liouville situation.
