  The article presents a compact review of the analytical results (2002-2009)
in the study of the system describing the motion of a top in two constant
fields. The Liouville integrability of this system under certain condition of
the Kowalevski type was established by A.G.Reyman and M.A.Semenov-Tian-Shansky.
We present some geometrical foundations of finding separations of variables.
Two systems of local planar coordinates are introduced leading to separation of
variables for two subsystems with two degrees of freedom in the dynamics of the
generalized Kowalevski top.
