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Due to Poinsot's theorem, the motion of a rigid body about a fixed point isrepresented as rolling without slipping of the moving hodograph of the angularvelocity over the fixed one. If the moving hodograph is a closed curve,visualization of motion is obtained by the method of P.V.Kharlamov. For anarbitrary motion in an integrable problem with an axially symmetric force fieldthe moving hodograph densely fills some two-dimensional surface and the fixedone fills a three-dimensional surface. In this paper, we consider theirreducible integrable case in which both hodographs are two-frequency curves.We obtain the equations of bearing surfaces, illustrate the main types of thesurfaces. We propose a method of the so-called non-straight geometricinterpretation representing the motion of a body as a superposition of twoperiodic motions.

On the geometry of motions in one integrable problem of the rigid body dynamics