Integrable Systems, Poisson Pencils, and Hyperelliptic Lax Pairs
Author(s): Fedorov Y. N.
In the given paper we consider a new Lax pair for the multidimensional Manakov system on the Lie algebra $so(m)$ with a spectral parameter defined on a certain unramified covering of a hyperelliptic curve. An analogous $L$–$A$ pair for the Clebsch–Perelomov system on the Lie algebra $e(n)$ can be indicated.
In addition, the hyperelliptic Lax pair enables us to obtain the multidimensional generalizations of the classical integrable Steklov–Lyapunov systems in the problem of a rigid body motion in an ideal fluid. The latter is known to be a Hamiltonian system on the algebra $e(3)$. It turns out that these generalized systems are defined not on the algebra $e(n)$, as one might expect, but on a certain product $so(m)+so(m)$. A proof of the integrability of the systems is based on the method proposed in [1].
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