  We derive a differential equation that is regular at the collision of two
equal-mass bodies with attractive interaction in the relativistic
action-at-a-distance electrodynamics. Our method uses the energy constant
related to the Poincar\'{e} invariance of the theory to motivate the
regularizing coordinate transformation and to remove infinities from the
equation of motion. The collision orbits are calculated numerically using the
regular equation adapted in a self-consistent minimization method (a stable
numerical method that chooses only nonrunaway solutions). This dynamical system
appeared 100 years ago as a time-symmetric relativistic motion and aquired the
status of electrodynamics in the 1940's by the works of Dirac, Wheeler and
Feynman. We outline the method with an emphasis on the physics of this complex
conservative dynamical system.
