  The properties of radiation created by a classical ultrarelativistic scalar
charged particle in a constant homogeneous crossed electromagnetic field are
described both analytically and numerically with radiation reaction taken into
account in the form of the Landau-Lifshitz equation. The total radiation
naturally falls into two parts: the radiation formed at the entrance point of a
particle into the crossed field (the synchrotron entrance radiation), and the
radiation coming from the late-time asymptotics of a particle motion (the
de-excited radiation). The synchrotron entrance radiation resembles, although
does not coincide with, the ultrarelativistic limit of the synchrotron
radiation: its distribution over energies and angles possesses almost the same
properties. The de-excited radiation is soft, not concentrated in the plane of
motion of a charged particle, and almost completely circularly polarized. The
photon energy delivering the maximum to its spectral angular distribution
decreases with increasing the initial energy of a charged particle, while the
maximum value of this distribution remains the same at the fixed observation
angle. The ultraviolet and infrared asymptotics of the total radiation are also
described.
