Bifurcation diagrams of the integral mappings of a top with singular symmetry

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In general case, a Hamiltonian system with three degrees of freedom describing the motion of a rigid body in two constant fields does not admit any symmetry groups. H.Yehia has found conditions under which the equations of motion of the Kovelevskaya type gyrostat in such a field have, in addition to the energy integral, an integral linear with respect to angular velocities. Later it was noticed that this integral under the conditions of Yehia exists for arbitrary axially symmetric gyrostat having the centers of the fields applications in the equatorial plane. The corresponding natural mechanical system has an S¹-symmetry which has a non-empty set of fixed points. Smale's program of topological analysis can be fulfilled with some modifications considering the singularity of the symmetry action. We construct the bifurcation diagrams of the momentum map for a family of systems with a singular symmetry and investigate the dependency on one essential parameter expressing the ratio of the axial and equatorial inertia moments.