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Compatible Poisson brackets on Lie algebras. (English. Russian original) Zbl 1042.37041
Math. Notes 72, No. 1, 10-30 (2002); translation from Mat. Zametki 72, No. 1, 11-34 (2002).
The authors discuss the relationship between the representation of an integrable system via a Lax pair with a spectral parameter and the existence of a bi-Hamiltonian representation. After reviewing some well-known methods of producing bi-Hamiltonian systems, the authors state the following result: Let $$\{~,~\}_{\lambda}$$ be a family of Poisson brackets on a linear vector space and $$v$$ a vector field, such that: (i) Almost all brackets in the family are isomorphic to the dual of a semisimple Lie algebra, and (ii) $$v$$ is Hamiltonian with respect to all brackets from the family. Then $$v$$ admits a Lax representation with a sprectral parameter.
The authors illustrate this result by considering some well-known examples of completely integrable systems: geodesic flows of left-invariant metrics on Lie groups, the Zhukovskii-Volterra system describing the inertial motion of a balanced gyroscope, the Kowaleski top, etc.

MSC:
 37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests 17B63 Poisson algebras 53D17 Poisson manifolds; Poisson groupoids and algebroids 37K30 Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with infinite-dimensional Lie algebras and other algebraic structures
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