Complete integrability of the equations of motion of a spatial pendulum in a medium flow with rotational derivatives of the torque produced by the medium taken into account

We construct a nonlinearmodel of the mediumaction on a rigid body taking into account the dependence of the force arm on the reduced angular velocity of the body. In this case, the moment of the action force itself is also a function of the angle of attack. As experimental data processing for the motion of homogeneous circular cylinders in water has shown, it is necessary to take these facts into account in modeling.Studying the model of interaction between the spatial pendulum and the medium, we found a new case of complete integrability in elementary functions. This allowed us to find several qualitative analogies between the motions of bodies that are free in the resisting environment and the oscillations of bodies partially fixed in the homogeneous flow of incoming medium.