Compatible Poisson brackets on Lie algebras (Q1810156)

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.