The linear combinations Λ$_{i}$ of the polarization operator components are calculated in [4] in one-loop approximation, the calculational details being presented in [5] on the basis of [38]. The latter reference as well as [6, 7] contains also calculations of alternative sets of four scalar coefficient functions of an appropriate set of basic matrices in terms of which the polarization operator may be expressed. However, the set of functions Λ$_{i}$ is preferred to them all, because the eigenvalues are given the simplest in their terms. In this Appendix we write the linear in electric field correction into invariant function Λ$_{3}$ for the case, where an electric field, much smaller than the magnetic field, is added parallel to the latter (this wording refers to the special Lorentz frame). The linear part of Λ$_{3}$ defines the leading contribution into the mixing of photon eigenmodes in a magnetic field due to the perturbation introduced by the electric field. We present it here as a result of one-loop calculations of quantum electrodynamics in external magnetic field. Using (11) and notations (7) the expansion (15) of the slightly perturbed eigenvectors over the eigenvectors in a magnetic field alone becomes Appendix 2 ♭ (2) µ = − (Λ$_{2}$ − Λ$_{4}$) ( k $^{2}$B2 + kF $^{2}$k ) 1 / 2 ♭ (2) µ ∣ G =0 + ( G (Λ$_{2}$ − Λ$_{4}$) B $^{2}$( k $^{2}$B2 + kF $^{2}$k ) 1 / 2 − Λ$_{3}$ ( − kF $^{2}$k ) 1 / 2 ) ♭ (3) µ ∣ G =0 , ♭ (3) µ = (Λ$_{2}$ − Λ$_{4}$) ( − kF $^{2}$k ) 1 / 2 ♭ (3) µ ∣ G =0 + ( G (Λ$_{2}$ − Λ$_{4}$) B $^{2}$( − kF $^{2}$k ) 1 2 − Λ$_{3}$ ( k $^{2}$B2 + kF $^{2}$k ) 1 2 ♭ (2) µ G =0 . (56) It is understood that Λ$_{2}$ − Λ$_{4}$ in the right-hand side are taken at G = 0. In writing these equations we took into account that Λ$_{2}$$_{,}$$_{4}$ are even, and Λ$_{3}$ is an odd functions of G . It is seen from (12) that at G = 0 the quantity − Λ$_{4}$ is the polarization operator eigenvalue κ 2 in a magnetic field. Analogously, − Λ$_{2}$ = κ $_{3}$. These quantities in the one-loop approxi- mation are known [4, 14]. Now we shall write Λ$_{3}$ in the same approximation by calculating the G = 0 limit of the corresponding expression from [4]. Λ$_{3}$ = − α 4 π$^{G}$ F ( − kF $^{2}$k 2 F ) 1 2 ( − k ˜ F $^{2}$k 2 F ) 1 2 ∫ ∞ 0 τ d τ sinh 2 τ ∫ 1 − 1 d η (1 − η ) $^{2}$sinh 2 τ (1 + η ) 2 × × exp { kF $^{2}$k 2 F sinh((1 + η ) / 2) τ ) sinh((1 − η ) τ / 2) sinh( ef τ ) − k ˜ F $^{2}$k 2 F 1 − η 2 4 ef τ − m 2 $_{e}$τ ef } . (57) 27