Integrable Systems, Poisson Pencils, and Hyperelliptic Lax Pairs

In the modern approach to integrable Hamiltonian systems, their representation in the Lax form (the Lax pair or the $L-A$ pair) plays a key role. Such a representation also makes it possible to construct and solve multi-dimensional integrable generalizations of various problems of dynamics. The best known examples are the generalizations of Euler's and Clebsch's classical systems in the rigid body dynamics, whose Lax pairs were found by Manakov [10] and Perelomov [12]. These Lax pairs include an additional (spectral) parameter defined on the compactified complex plane or an elliptic curve (Riemann surface of genus one). Until now there were no examples of $L-A$ pairs representing physical systems with a spectral parameter running through an algebraic curve of genus more than one (the conditions for the existence of such Lax pairs were studied in [11]).
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].