  The reduced system in the Clebsch problem of the motion of a rigid body in
fluid treated as the motion of a rigid body about its fixed mass center in a
central Newtonian field with zero value of the area integral is a completely
integrable natural mechanical system with two degrees of freedom. We find out
the phase topology of this system including constructing the bifurcation set of
two integrals quadratic with respect to velocities and describing all types of
bifurcations of the integral tori. We establish the topology of all singular
integral surfaces. Using the contemporary language, one can say that this is
the first description of the bifurcations of the types $B$ ($T^2\to 2T^2$) and
$C_2$ ($2T^2 \to 2T^2$), for the latter see also arXiv:1408.4548. This
investigation includes the description of the foliation into integral surfaces
of an invariant neighborhood of a saddle type singularity having two
equilibriums on the connected critical integral surface.
