Complete separability of the Hamilton–Jacobi equation for the charged particle orbits in a Liénard–Wiechert field

We classify all orthogonal coordinate systems in M4, allowing complete additively separated solutions of the Hamilton–Jacobi equation for a charged test particle in the Liénard–Wiechert field generated by any possible given motion of a point-charge Q. We prove that only the Cavendish–Coulomb field, corresponding to the uniform motion of Q, admits separation of variables, precisely in cylindrical spherical and cylindrical conical-spherical coordinates. We show also that for some fields, the test particle with motion constrained into certain planes admits complete orthogonal separation, and we determine the separable coordinates.