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Canonical forms for bihamiltonian systems. (English) Zbl 0823.58019
Babelon, Olivier (ed.) et al., Integrable systems: the Verdier memorial conference. Actes du colloque international de Luminy, France, July 1-5, 1991. Boston, MA: Birkhäuser. Prog. Math. 115, 239-249 (1993).
A system of differential equations is called bi-Hamiltonian if it can be written in Hamiltonian form in two distinct ways: $$dx/dt = J_ 1\nabla H_ 1= J_ 2\nabla H_ 0$$. The bi-Hamiltonian structure determined by $$J_ 1$$, $$J_ 2$$, is compatible if the sum $$J_ 2+ J_ 2$$ is also Hamiltonian. The bi-Hamiltonian structure is nondegenerate if the operator $$J_ 1$$ is nonsingular.
First the author reviews some results on the canonical forms for compatible, nondegenerate complex-analytic bi-Hamiltonian systems. Secondly, the author briefly describes the results of Brouzet and Fernandes on integrable Hamiltonian systems without bi-Hamiltonian structures.
For the entire collection see [Zbl 0807.00017].
Reviewer: Li Jibin (Kunming)

MSC:
 37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests 37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.) 35Q51 Soliton equations