Within the framework of the planar model of the rotor mounted on anisotropic elastic-viscous supports and balanced by a passive auto-balancer, conditions for the occurrence of auto-balancing were analytically determined.An empirical criterion for stability of the main motion was applied. It was found that depending on the forces of viscous resistance in supports, the rotor has one or three critical speeds. These speeds are between two natural frequencies of rotor oscillation in absence of resistance forces in supports. Auto-balancing, respectively, occurs when the single critical speed is exceeded or between the first and the second and above the third critical speeds.At low forces of viscous resistance, the rotor has three critical speeds. The first and the third critical speeds coincide with two natural frequencies of rotor oscillation in absence of resistance forces in supports. The second critical speed is between the first two. An additional (second) critical speed appears when the auto-balancer is mounted on the rotor. In the transition of this speed the behavior of the auto-balancer changes: the auto-balancer reduces the rotor imbalance at slightly lower rotor speeds and increases it at somewhat higher speeds.At finite forces of viscous resistance in supports, depending on the magnitude of these forces, the rotor has one or three critical speeds.At large forces of viscous resistance in supports, the rotor has one critical speed. Depending on the relationship between the coefficients of the forces of viscous resistance, this speed is closer to the smallest or the largest natural frequency of the rotor oscillation.The results obtained were confirmed by computational experiments. It was established that the criterion correctly describes the qualitative behavior of the rotor – auto-balancer system: it determines the number of critical speeds and the region of the auto-balancing onset. Accuracy of determining critical speeds (the boundaries of the regions of auto-balancing onset) increases with:– reduction of the auto-balancer mass with respect to the rotor mass;– an increase in forces of viscous resistance to the motion of correction weights