The energy method of studying rotor dynamics has been modernized. The method is applicable to the rotors mounted on isotropic elastic-viscous supports when bodies are attached to the rotors and relative motion of these bodies is prevented by elastic and viscous forces. The method is designed to search for steady motions and determine conditions of their existence as well as assess stability of the rotor system. Relative motions of the attached bodies cease at steady motions and the system rotates as a single whole around the axis of rotation formed by supports. Effectiveness of the method was illustrated by an example of a flat model of a rotor and an auto-balancer with many loads in the form of balls, rollers or pendulums. It has been established that the system has family of main motions (the rotor is balanced at them) both with and without damping in supports at a sufficient balancing capacity of the auto-balancer. In the absence of damping in supports, the system has: ‒ isolated secondary motions at which the rotor is unbalanced and centers of mass of the loads are deflected to the side of imbalance or in the opposite direction if there is unbalance of the rotor; ‒ one-parameter families of secondary motions at which the centers of mass of the loads lie on one straight line in the absence of unbalance of the rotor. In the presence of damping in supports: ‒ the system has isolated secondary motions at which the centers of mass of the loads lie on one straight line and this straight line forms an angle with the imbalance vector depending on the rotor speed in the presence of the rotor imbalance; ‒ there are no secondary motions in the absence of the rotor imbalance. The secondary motions and domains of their existence do not depend on the angular velocity of the rotor in the absence of damping in supports but they depend on the angular velocity of the rotor in the presence of the rotor imbalance. Both in the presence and in the absence of damping in supports: ‒ only the secondary motion at which total imbalance of the rotor and loads is greatest can be stable at sub-resonant rotor speeds; ‒ only a family of main motions can be stable at super-resonant rotor speeds.