Stationary solutions in a model three-body problem

Two visco-elastic bodies (deformable spheres) are considered which interact with each other and move in quasi-circular orbits in the attractive force field of a fixed centre - a heavy point mass. Their axes of rotation are perpendicular to their orbital plane. Stationary solutions of the evolutionary equations of motion are found. In one particular case, they extend solutions of the restricted circular three-body problem corresponding to two collinear libration points. All three bodies are located along a straight line. This implies synchronization of motion of the barycentre of the two visco-elastic bodies relative to the attracting centre with their orbital motion relative to the barycentre in a 1:1 resonance. The rotation of the two bodies relative to their own centres of mass takes place in such a way that the bodies “view” the attracting centre and each other from the same side, i.e., they are synchronized in a 1:1 resonance with their orbital motion. Instability of stationary solutions is analytically proven.