satisfactorily considered in the case of E$_{‖}$ = 0. Also it is not clear beforehand how will the processes that depend on the resonance behave under inclusion of even small electric field. On the other hand, extremely strong electric fields with the strength as high as ∼ (10 ÷ 10 $^{2}$) E$_{0}$ are predicted to exist at the surface of bare strange stars that are entirely made of deconfined quarks, where E$_{0}$ = m 2 $_{e}$/e ≃ 1 . 3 × 10 16 V/cm is the characteristic electric field value [32]. These electric fields are directed perpendicular to the stellar surface and prevent ultra-relativistic electrons of quark matter from their escape to infinity. The surface magnetic fields are expected to be more or less the same for neutron and strange stars (from ≲ 10 9 G to ∼ 10 15 G or even higher), and therefore, at the surface of strange stars the ratio E$_{‖}$/B may be both ≪ 1 and ≫ 1, i.e., it may be practically arbitrary. We, therefore, feel that the time has come to renew the study of the general combination of constant electric and magnetic field as a strong external background to smaller electro- magnetic fields on the base of the recent-years experience acquired when considering the magnetic-like field. In this paper we are elaborating consequences of the general structure of the polarization operator, established in [4], for the propagation and polarization of photons and massive vector excitations of the vacuum in the presence of a constant background field with the both field invariants different from zero. Also some observations are made depending on the one-loop approximation. These are exiled to Appendix 2. In Section II we review properties of the polarization operator and of its three excitation eigenmodes that follow from its diagonal representation, derived in [4] from the fundamental principles in an approximation- and model-independent way. We present a kinematical or- thogonal basis and the decomposition of the eigenvectors of the polarization operator over it. The coefficients in this decomposition are expressed in terms of four linear invariant combi- nations, called Λ$_{1}$$_{,}$$_{2}$$_{,}$$_{4}$$_{,}$$_{5}$, of the polarization operator components. We also present the simpler form of this decomposition valid in the limit, where the mixing between the basic vectors is small, like for small admixture of an electric field to a large magnetic one, and under special kinematical conditions. We establish that under this admixture the distance between dis- persion curves of two modes that can carry massless excitations (photons) increases in the transparency domain as a consequence of Hermiticity, thus leading to increasing the bire- fringence and strengthening Adler’s kinematical bans [16] for photon splitting in a magnetic field. We discuss how Adler’s CP-selection rules are modified in the case of the general field 6