Periodic motions of a reversible mechanical system under perturbations of general form that do not
preserve system’s reversibility are investigated. The conditions for the existence of periodic motions of
a perturbed system that turn at a zero value of a small parameter into symmetrical periodic motions not
degenerating into equilibriua are found. Both autonomous and periodic perturbations are considered.
Amplitude equations, whose simple roots correspond to periodic solutions of the perturbed system, are
derived for the systems investigated. As a result, cycles are found for the autonomous system, and isolated
periodic motions are found for the periodic system. Both a separate system and a model containing
coupled subsystems are investigated. The characteristic exponents ofthe cycles are calculated. Itis proved
that any oscillation of a system that can be described by Lagrange’s equations with positional forces is
symmetrical and belongs to a family. It is also deduced that for the realization of a cycle or, in the case
of non-autonomous forces, an isolated periodic motion, velocity-dependent forces of definite structure
are needed. The results obtained are applied to the problem of the oscillations of a satellite under the
action of gravitational and aerodynamic torques. The existence of asymmetrical isolated oscillations in a
low-eccentricity elliptical orbit is established