In this paper, the conditions of nonregular precession with a constant ratio of the velocities
of precession and proper rotation for a gyrostat in the superposition of two homogeneous and
one axisymmetric field are obtained. The case where the gyrostat has axial dynamical symmetry
and the proper rotation axis coincides with the body’s symmetry axis is singled out. It is shown
that in the case where the gyrostatic momentum is collinear to the symmetry axis, nonregular
precession is possible with a precession velocity equal to, twice as large as, or twice as small
as the proper rotation velocity. In each of these cases, the condition expressing the ratio of
the axial and equatorial inertia moments of the body in terms of the nutation angle coincides
with the corresponding condition obtained earlier for the nonregular precession of a solid in
three homogeneous fields. In the particular case of the gyrostat’s spherical symmetry, when the
precession speed is half or twice as large as its proper rotation speed, the cosine of the nutation
angle is equal to one fourth; at equal speeds, the nutation angle should be equal to sixty degrees.
The sets of admissible positions of the forces’ centers for the general case of nonorthogonal
fields are found. The precession of a gyrostat whose gyrostatic momentum is deflected from the
symmetry axis is considered. The possibility of nonregular precession is shown for the case where
the precession velocity is twice as large as the proper rotation velocity. The solution is expressed
in terms of elementary functions. The rotation of the gyrostat is either periodic or the rotation
velocity tends to zero and the carrier body of the gyrostat approaches the equilibrium position.
