  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 $S^1$-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.
