  We derive the nonlinear equations governing the dynamics of three-dimensional
(3D) disturbances in a nonuniform rotating self-gravitating fluid under the
assumption that the characteristic frequencies of disturbances are small
compared to the rotation frequency. Analytical solutions of these equations are
found in the form of the 3D vortex dipole solitons. The method for obtaining
these solutions is based on the well-known Larichev-Reznik procedure for
finding two-dimensional nonlinear dipole vortex solutions in the physics of
atmospheres of rotating planets. In addition to the basic 3D x-antisymmetric
part (carrier), the solution may also contain radially symmetric (monopole)
or/and antisymmetric along the rotation axis (z-axis) parts with arbitrary
amplitudes, but these superimposed parts cannot exist without the basic part.
The 3D vortex soliton without the superimposed parts is extremely stable. It
moves without distortion and retains its shape even in the presence of an
initial noise disturbance. The solitons with parts that are radially symmetric
or/and z-antisymmetric turn out to be unstable, although at sufficiently small
amplitudes of these superimposed parts, the soliton retains its shape for a
very long time.
