Fedorov, Yu. Integrable systems, Poisson pencils, and hyperelliptic Lax pairs. (English. Russian original) Zbl 0985.37060 J. Math. Sci., New York 94, No. 4, 1501-1511 (1999); translation from Zap. Nauchn. Semin. POMI 235, 87-103 (1996). From the author’s introduction: We consider a new Lax pair for the multidimensional Manakov system on the Lie algebra \(\text{so}(m)\) 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 \(\text{so}(m)+ \text{so}(m)\). A proof of the integrability of the systems is based on the method proposed in [A. Bolsinov, Acta Appl. Math. 24, 253-274 (1991; Zbl 0746.58035)]. MSC: 37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests 37N05 Dynamical systems in classical and celestial mechanics 70H06 Completely integrable systems and methods of integration for problems in Hamiltonian and Lagrangian mechanics 70E40 Integrable cases of motion in rigid body dynamics 70E45 Higher-dimensional generalizations in rigid body dynamics Keywords:integrable Hamiltonian systems; Poisson brackets; Lax pair; multidimensional Manakov system; Lie algebra so\((m)\); L-A pair; Clebsch-Perelomov system; hyperelliptic Lax; integrable Steklov-Lyapunov systems Citations:Zbl 0924.00015; Zbl 0746.58035 × Cite Format Result Cite Review PDF Full Text: DOI EuDML References: [1] A. Bolsinov, ”Commutative families of functions related to consistent Poisson brackets,”Acta Applicandae Mathematicae,24, 253–274 (1991). · Zbl 0746.58035 · doi:10.1007/BF00047046 [2] A. Bolsinov, and Yu. Fedorov, ”Multidimensional integrable generalizations of the Steklov-Lyapunov systems,”Vest. MGU, Ser. Mat. Mekh.,6, 53–56 (1992). · Zbl 0774.70008 [3] B. A. Dubrovin, I. M. Krichever, and S. P. Novikov,Integrable Systems. I.Itogi Nauki i Tekhniki. Sovremennye Problemy Matematiki. Fund. Naprav. [in Russian], Vol. f 4, VINITI, Moscow (1985). · Zbl 0780.58019 [4] Yu. Fedorov, ”Lax representations with a spectral parameter on a covering of a hyperelliptic curve,”Nat. Zam.,54, No. 1, 94–109 (1993). [5] F. Frahm, ”Uber gewisse Differentialgleichungen,”Math. Ann.,8, 35–44 (1974). · JFM 06.0226.02 · doi:10.1007/BF01970877 [6] L. Heine, ”Geodesic flow on so (4) and abelian surfaces,”Math. Ann.,263, 435–472 (1983). · Zbl 0521.58042 · doi:10.1007/BF01457053 [7] F. Kötter, ”Die von Steklow und Lyapunow entdeckten integralen Falle der Bewegung eines starren Körpers in einer Flüssigkeit,” Sitzungberichte der Königlich preussischen Academie der Wissenschaften zu Berlin,6, 79–87 (1900). [8] A. M. Lyapunov, ”A new integrable case of the equations of motion of a rigid body in a fluid,”Fortshtitte der Mathem., Bd. 25,1501 (1897). [9] F. Magri, ”A simple model of the integrable Hamiltonian equation,”J. Math. Phys., 1156–1162 (1978). · Zbl 0383.35065 [10] S. V. Manakov, ”Remarks on the integtals of the Euler equations of then-dimensional heavy top,”Func. Anal. Appl.,10, 93–94 (1976). [11] A. V. Mikhailov and V. E. Zakharov, ”The method of the inverse spectral problem with a spectral parameter on an algebraic curve,”Funct. Anal. Appl.,4, 1–6 (1983). [12] A. Perelomov, ”Some remarks on integrability of equations describing a rigid body movement in an ideal liquid,”Funct. Anal. Appl.,15, No. 2, 83–85 (1981). · Zbl 0495.70016 · doi:10.1007/BF01082293 [13] V. Rubanovsky, ”Integrable cases in the problem of a rigid body motion in a fluid,”Dokl. Akad. Nauk USSR,180, 556–559 (1968). [14] V. Steklov, ”Über die Bewegung eines festen Körpers in einer Flüssigkeit,”Math. Ann.,42, 273–274 (1893). · JFM 25.1499.01 · doi:10.1007/BF01444182 [15] V. Volterra, ”Sur la theorie des variations des latitudes,”Acta Math.,22, 201–357 (1899). · JFM 29.0650.01 · doi:10.1007/BF02417877 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.