Solution to Maxwell-Lorentz equations I am trying, without success, to find an example (preferably simple) of solution for the Maxwell-Lorentz equations, i.e., the coupled system of Maxwell equations + dynamics of a charged particle given by Lorentz force. Say we have a (for simplicity, non-relativistic) particle of mass m, charge q, position $\vec x$ and velocity $\vec v$, then the Lorentz force will give
$$m   \vec x''(t) = q ( \vec E (\vec x(t),t) +  \vec v(t)  \times  \vec B (\vec x(t),t ))$$
Is there any system for which we can exhibit at some instant $t_0$ the 'state' of the system $(  \vec E(\vec r,t_0),\vec B(\vec r,t_0) ,\vec x(t_0),\vec v(t_0))$?
Standard textbooks seems not to consider solutions of coupled Maxwell-Lorentz equations, the only one I didn't check is Jackson's, because I don't have a copy with me.
 A: The Maxwell-Lorentz equations for point-like charged particles are meaningless. This is well-known since the beginnig of the 20th century. Older textbooks (like that of Becker)  written between the two world wars discuss it in all details. The devil lies in the self-interaction. A hand-made correction, excluding from the Lorentz force the field produced by the particle itself has still survived. Ignoring all magnetic forces this approach leads to the Coulomb Hamiltonian  used also  in the non-relativistic quantum mechanics,  where the Coulomb terms i=j are just omitted. 
Actually, one should not even teach the electrodynamics of point-like classical particles, since it is basically wrong. Has neither Lagrangian nor Hamiltonian formulation. A consistent formulation of the electrodynamics of charged particles may be formulated only in the frame of the field theory followed by a quantization. 
I recommend You the recent pedagogical arXiv preprint ( a future chapter of a textbook in preparation):
A field-theoretical approach to non-relativistic QED.
by  Ladislaus Alexander Bányai and Mircea Bundaru,
arXiv:1907.13053v1 [quant-ph] 30 Jul 2019 
A: May be, one my construct such theories (the only ones I am aware, introduce a charge distribution for the classical particles) , but these  are not the Lorentz-Maxwell theory. The whole development of quantum mechanics started from this one  and has been developed along his lines to get finally the QED. Any other variants left no traces in the development of physics. 
