contrast to the earlier publications on the similar subject, where only classical approach was adopted (see, e.g., [@Altschul], and also [@Montemayor_Urrutia], [@Ellis_Mavromatos_Sakharov]), we shall carry out consistent quantum consideration of the phenomena, including [*both*]{} the motion of an electron [*and*]{} its radiation. We take Lorentz invariance violation into account exactly, i.e. avoiding (at the stage of quantization) the perturbation theory approximation and considering $ b^{0} $ as a given classical external field in addition to the magnetic field background. We obtain a system of exact solutions of the modified Dirac equation for an electron with a vacuum magnetic moment in a constant magnetic field, which is necessary for the quantum description of the fermion field in the Furry picture [@Furry]. Employing the methods of QED, with the use of these solutions, we calculate the asymptotic expressions for the spectral-angular distribution of the electromagnetic radiation in the weak-field limit which incorporate yet [*all*]{} the quantum corrections arising, and we pay special attention to the radiation properties caused by the Lorentz invariance violation assumed in the theory.

Throughout the work, the natural units $ \hbar = c = 1 $ are adopted.

Motivations {#theory}
===========

In this section, let us recall some 