Integrable Euler equations on Lie algebras arising in problems of mathematical physics

Complete integrability in the sense of Liouville is established for the rotation of an arbitrary rigid body about a fixed point in a Newtonian field with an arbitrary homogeneous quadratic potential. Explicit formulas, which express the angular velocity of the rigid body rotation in terms of theta functions on Riemannian surfaces, are obtained. A series of cases is found in which the Euler equations on the Lie algebra $\operatorname{SO}(4)$ are integrable. A model of pulsar rotation, the dynamics of which are described by Euler equations on the Lie algebra $\operatorname{SO}(3)\oplus E_3$, is investigated.
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