The conditions for the occurrence of auto-balancing when balancing a flexible axisymmetric rotor by any number of passive auto-balancers of any type are determined. The problem is actual for the high-speed rotors working at supercritical speeds (rotors of aircraft engines, gas turbine engines of power plants, etc.). The empirical criterion for the occurrence of auto-balancing is applied. Transformations were carried out on the example of the flexible axisymmetric rotor of constant section on two rigid hinge supports. The findings are applicable to rotors with another type of fixing. It is established that auto-balancing of the rotor by n passive auto-balancers located in different correction planes is possible only if the rotor speed exceeds the n-th critical speed. The number of auto-balancers can be arbitrary. Between the critical rotor speeds, additional critical speeds appear. Auto-balancing occurs whenever the rotor passes a critical speed and disappears whenever the rotor passes an additional critical speed. If n auto-balancers are located in the n nodes of the rotor flexural (n+1)-th mode, the j·n-th additional critical rotor speed matches with the j(n+1)-th critical speed, /j=1, 2, 3,…/. When balancing the flexible rotor between the n-th and (n+1)-th critical speeds, such number and placement of auto-balancers are optimum. Auto-balancers at the same time balance the first n distributed modal unbalances and do not respond to the (n+1)-th ones. The additional critical speeds are due to the installation of the auto-balancers on the rotor. Upon transition to them, the behavior of auto-balancers changes. At slightly lower rotor speeds, the auto-balancers reduce the rotor unbalance, and at slightly higher ones – increase it.