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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: $dx/dt = J\sb 1\nabla H\sb 1= J\sb 2\nabla H\sb 0$. The bi-Hamiltonian structure determined by $J\sb 1$, $J\sb 2$, is compatible if the sum $J\sb 2+ J\sb 2$ is also Hamiltonian. The bi-Hamiltonian structure is nondegenerate if the operator $J\sb 1$ 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].

37J35Completely integrable systems, topological structure of phase space, integration methods
37K10Completely integrable systems, integrability tests, bi-Hamiltonian structures, hierarchies
35Q51Soliton-like equations