  We present a review of the results obtained during the last fifty years in
the problem of the motion of a heavy gyrostat under the conditions of the
Kowalevski type. Hamad M. Yehia in 1986 has proved that the problem is complete
integrable. Since then, a lot of works were devoted to different aspects of
integrating this problem and of its topological investigation. The main idea of
this work is to prove strictly all the facts of qualitative analysis and to fix
different mistakes and inaccuracies. The work was supported by the grants of
the RFBR No. 10-01-00043, 10-01-97001, 13-01-97025, and 14-01-00119.
