Relativistic charge in plane wave

It is proved that the system of Maxwell–Lorentz equations in $z$-presentation has many solutions. Determination of the energetic condition of the charge is reduced to analysis of the parametric system as a Kepler problem. The extreme conditions and configuration of the electromagnetic wave field determine the value of the reflection parameter, which changes the efficiency of electron-wave interaction and dynamics of the charge's motion. A visual geometric interpretation of possible energetic state of the electron is given. We present expressions that determine the change of the charge's energy and coordinates as functions of the length of interaction space and laboratory time. Four possible regimes of the electron energy change are found. We obtain a functional connection of the initial impulses with the electromagnetic wave amplitude. The exact theoretical solution of electron motion relativistic equations is constructed in the form of a uniformly converging Fourier series.