Cases of integrability corresponding to the motion of a pendulum on the two-dimensional plane

We systematize some results on the study of the equations of plane-parallel motion of symmetric fixed rigid bodies–pendulums located in a nonconservative force fields. The form of these equations is taken from the dynamics of real fixed rigid bodies placed in a homogeneous flow of a medium. In parallel, we study the problem of a plane-parallel motion of a free rigid body also located in a similar force fields. Herewith, this free rigid body is influenced by a nonconservative tracing force; under action of this force, either the magnitude of the velocity of some characteristic point of the body remains constant, which means that the system possesses a nonintegrable servo constraint, or the center of mass of the body moves rectilinearly and uniformly; this means that there exists a nonconservative couple of forces in the system. Key–Words: Rigid body, Pendulum, Resisting Medium, Dynamical Systems With Variable Dissipation, Integrability