Summary: The influence of anisotropy of elastic bearings on forced oscillations of a rotor with the static and moment unbalance is studied for the cases of its fastening on a rigid shaft and on a flexible one. The rotor with four degrees of freedom is considered. It is suggested that the shaft is fixed in linear elastic nonisotropic bearings. The differential equations of rotation of the rotor are written in complex variables, and an exact solution to the equation system is found that corresponds to the elliptical synchronous precession. The exact solution is a sum of two vectors, one of which parametrizes a forward precession, while another parametrizes a reverse precession. Amplitude-frequency characteristics of forward and reverse precessions and elliptical trajectories of the rotor axis ends are constructed. It is shown that, in case of nonisotropic bearings, both the forward and reverse precession, as well as the axis motion of nonsimple type (when its one end is moving forward, while another is moving in the reverse direction), can take place. The influence of anisotropy of elastic bearings also manifests itself by change in critical frequencies towards their reduction and by arising of additional critical frequencies in the bottom part of the spectrum, which significantly complicates dynamics of the high-speed rotor at the moment when it reaches the working angular speed.