  A system of reduced equations is proposed for the electron motion in the
strongly-radiation dominated regime for an arbitrary electromagnetic field
configuration. The developed approach is used to analyze various scenarios of
an electron dynamics in the strongly-radiation dominated regime: motion in
rotating electric and magnetic fields, longitudinal acceleration in a plane
wave and in a plasma wakefield. The obtained results show that the developed
approach is able to describe features of the electron dynamics, which are
essential to a certain scenario, but which could not be captured in the
framework of the original radiation-free approximation [A. S. Samsonov et al.,
Phys. Rev. A 98, 053858 (2018); A. Gonoskov and M. Marklund, Phys. Plasmas 25,
093109 (2018)]. The results are verified by numerical integration of
non-reduced motion equations with account of radiation reaction in both
semi-classical and fully quantum cases.
