Almost circular orbits in classical action-at-a-distance electrodynamics

The motion of two fully relativistic classical spinless point particles interacting electromagnetically is studied in the special case of almost circular orbits. The equations of motion are difference-differential equations with half-retarded plus half-advanced Li\'enard-Wiechert potentials. As expected on both physical and mathematical grounds we find multiple stable solutions and conclude that ordinary (Newtonian) initial conditions are not sufficient to determine the trajectories. In addition to the stable solutions we find an infinity of divergent solutions. Alternatives for dealing with the extraneous solutions are discussed. The exact equations of motion can in certain limits be approximated by differential equations. Our solutions serve to delimit the range of applicability of the approximations.