Two problems of a torque-free body with a fixed point are considered. In both the body is rigid but for an internal rotor symmetric about its axis of rotation. In one the rotor is driven at a constant angular velocity, in the other the interaction torque between body and rotor has a zero component on the axis of relative rotation. The dynamical equations are shown to have exactly the same structure. A brief literature summary on the torque-free gyrostat is appended.

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