Kovalevskaya's solution [1] of the problem of motion of a heavy rigid body about a fixed point was generalized to the case of a double constant field in [2,3]. The corresponding Hamiltonian system has three degrees of freedom. The invariant four-dimensional submanifolds of the phase space were found in [2,4]. The case of [2] was studied in [5]. In this paper, we consider the case of [4].Consider a rigid body having a fixed point and satisfying the Kovalevskaya condition for the principal moments of inertia at the fixed point I = diag {2, 2, 1}. Suppose that the body is placed in a force field with the potential Address:

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In the general case the Hamiltonian system with three degrees of freedom describing the motion of a rigid body in two constant forces does not admit any symmetry groups. Yehia (1986 Mech. Res. Commun. 13 169-72) found conditions under which the equations of motion of the Kowalevski-type top have an integral linear in angular velocities in addition to the energy integral. Later it was noticed that such an integral exists for the same force field for any dynamically symmetric top with the center of force applications in the equatorial plane. Thus, the corresponding system is the natural mechanical system with S 1 -symmetry and Smale's program of topological analysis can be fulfilled. Here we construct the bifurcation diagrams of the momentum map for this system and present various types of diagrams depending on one physical parameter.

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Due to Poinsot's theorem, the motion of a rigid body about a fixed point is represented as rolling without slipping of the moving hodograph of the angular velocity 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 an arbitrary motion in an integrable problem with an axially symmetric force field the moving hodograph densely fills some two-dimensional surface and the fixed one fills a three-dimensional surface. In this paper, we consider the irreducible integrable case in which both hodographs are two-frequency curves. We obtain the equations of bearing surfaces, illustrate the main types of the surfaces. We propose a method of the so-called non-straight geometric interpretation representing the motion of a body as a superposition of two periodic motions.

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