  We present new exact solutions of the Landau-Lifshitz and higher-order
Landau-Lifshitz equations describing particle motion, with radiation reaction,
in intense electromagnetic fields. Through these solutions and others we
compare the phenomenological predictions of different equations in the context
of the conjectured `radiation-free direction' (RFD). We confirm analytically in
several cases that particle orbits predicted by the Landau-Lifshitz equation
indeed approach the RFD at extreme intensities, and give time-resolved signals
of this behaviour in radiation spectra.
