by a point electric charge placed in the background electromagnetic field. In Section V we establish relations to be obeyed by the four invariant combinations of the polarization operator components Λ$_{1}$$_{,}$$_{2}$$_{,}$$_{4}$$_{,}$$_{5}$ under special kinematical conditions that let the symmetry of the external field under rotations around its direction in the special frame and under Lorentz boost along this direction manifest itself as degeneracies of the polarization tensor, i.e. coincidences between pairs of its eigenvalues. Appendix 1 is technical. In Appendix 2 we present the one-loop approximation for the function Λ$_{3}$ in the limit of small electric admixture to the external magnetic field, G → 0, responsible for mixing eigenmodes in this limit. It has cyclotron resonances starting with the second threshold of electron-positron pair creation by a photon in a magnetic field. Therefore, the mixing does not affect the photon capture effect at the first threshold important for radiation formation in the pulsar magnetosphere. II. POLARIZATION OPERATOR, ITS EIGENVECTORS AND EIGENVALUES Before starting, technical conventions are in order. There are two field invariants F = 1 $_{4}$F$_{ρσ}$ F$_{ρσ}$ and G = 1 $_{4}$F$_{ρσ}$ ˜ F$_{ρσ}$ of the background fields and two Lorentz-scalar combinations kF $^{2}$k and k ˜ F $^{2}$k of the background field strength and momentum k$_{µ}$ of the elementary excitation, subject to the relation The dual field tensor is defined as ˜ F$_{ρσ}$ = i $_{2}$ϵ$_{ρσλ}$ κ F$_{λ}$ κ , where the completely antisymmetric unit tensor is fixed in such a way that ϵ$_{1234}$ = 1. We use the notations ( ˜ Fk )$_{µ}$ ≡ ˜ F$_{µτ}$k$_{τ}$ , ( Fk )$_{µ}$ ≡ F$_{µτ}$k$_{τ}$ , F 2 µν ≡ F$_{µτ}$F$_{τ ν}$ , ( F $^{2}$k )$_{µ}$ ≡ F 2 $_{µτ}$k$_{τ}$ , kF $^{2}$k ≡ k$_{µ}$F 2 $_{µτ}$k$_{τ}$ , k 2 ≡ k 2 + k 2 4 = k 2 − k 2 0 and are working in Euclidian metrics with the results analytically continued to Minkowsky space, hence we do not distinguish co- and contravariant indices. The scalar variables k ˜ F $^{2}$k 2 F − k 2 = kF $^{2}$k 2 F . (1) 8