Classification locale simultanée de deux formes symplectiques compatibles

Existence of conservation laws and characterization of recursion operators for completely integrable systems

“…More precisely, we have: Note. Recently, F. J. Turiel [17], using a completely different technique, obtained a nice generalization of this last result in the C ∞ case (the essential ideas appeared in a different context in his earlier paper [16]). His result agrees with our normal form in Theorem 3.4.…”

On a class of dynamical systems both quasi-bi-Hamiltonian and bi-Hamiltonian

“…The above form of the BH structure will be referred to as a canonical BH structure. We recall that the local model of a BH structure was studied in [5], where it was shown that a compatible pair (P 0 , P 1 ) of Poisson tensors (being P 0 invertible) admits a representation with P 0 constant and P 1 depending linearly on the coordinates; the Darboux-Nijenhuis coordinates are just a particular realization of this representation: the eigenvalues of N are the first n coordinates and the remaining ones are constructed by quadratures [4].…”

Bi-Hamiltonian structures and singularities of integrable systems

“…Being rewritten in analytical form, this condition means that the so-called Schouten bracket {{A, B}} of the Poisson structures A and B identically vanishes, which amounts to a rather non-trivial system of PDEs. A local description of compatible Poisson brackets can be found in (see [23][24][25]). …”