Bifurcations of the integral manifolds in the problem on motion of a heavy gyrostat

We study the topological structure of a common level surfaces of the first integrals in the problem on motion of a heavy gyrostat about a fixed point. We consider the special case when the gyrostatic momentum is collinear with the center-of-mass vector. With this supposition the axes of steady rotations can be directed only along generatrices of the Staude cone. We investigate the critical points of the effective potential, classify the bifurcation diagrams on the plane of constants of the first integrals and give a complete description of the topology of nonsingular integral manifolds of this problem.