On the motion of a multidimensional body with fixed point in a gravitational field

The problem of the motion of an $n$-dimensional solid body with a fixed point in a gravitational field is considered. More precisely, the integrable case of such motion determined by certain symmetry conditions of the body is considered. These conditions are obtained as a generalization of the conditions for the Lagrange case of the motion of a three-dimensional heavy gyroscope. For the $n$-dimensional Lagrange case the collection of first integrals presented in the paper is sufficient for complete integrability. The fact that the case considered provides an example of a completely integrable Hamiltonian system with a noncommutative algebra of first integrals is of interest.
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