Dynamics of an autorotating rigid body moving through the homogeneous atmosphere is studied. A sufficiently simple approach is used to modeling the aerodynamic forces and torques acting on the rigid body in an air stream. The pressure center is supposed to be at a fixed point in a body, and rotating torque is supposed to be a linear function of the angular velocity. The proposed method allows carrying out a sufficiently complete analysis of various motion modes. The main purpose of the work is the detailed analysis of specific stationary motion of an autorotational rigid body, the vertical motion: the uniform vertical descent of the body mass center and the uniform rotation of the body about vertically oriented longitudinal axis. The asymptotic stability of vertical motion is considered in three cases: for a spherical body, for an axial symmetric body, and for an arbitrary body. In the first case the second Liapunov's method was used. In the second case the conditions for asymptotic stability were obtained using the generalized Routh and Hurwitz criterion for polinomials with complex coefficients. In the third case the necessary and sufficient conditions for asymptotic stability were examined using Lienard and Chipart criterion. Several examples illustrate the derived results. It is shown how angular velocity of autorotation, aerodynamic coefficients, inertial and geometrical parameters of the body influence on the vertical motion stability.