polarization modes that cannot carry massless excitations. The reason why the magnetic field attracted so much attention was, of course, the dis- covery of extremely strong magnetic fields (up to ∼ 10 14 − 10 15 G) in the vicinity of many compact astronomical objects (soft gamma-ray repeaters, anomalous X-ray pulsars, and some radio pulsars) identified with rotating neutron stars [21]. Still stronger magnetic fields ( B ∼ 10 $^{16}$− 10 17 G) were predicted to exist at the surface of cosmological gamma-ray bursters if they are rotation-powered neutron stars similar to radio pulsars [22]. The remarkable fea- ture of these magnetic field strengths is that they are higher than the characteristic field value B$_{0}$ = m 2 $_{e}$c$^{3}$/e ℏ ≃ 4 . 4 × 10 13 G above which the nonlinearity of QED becomes actual, where m$_{e}$ and e are the electron mass and its charge, respectively. [Henceforth, we set ℏ = c = 1 and refer to the Heaviside-Lorentz system of units.] The compact astronomical objects identified with strongly magnetized neutron stars are powerful sources of electro- magnetic radiation, and therefore, propagation of photons in a strong magnetic field is one of the central problems in the theory of these objects. Correspondingly, these problems have been extensively studied aiming at applications to the pulsar physics (for a review, see [23]). In parallel, some more academic features of nonlinear electrodynamics in a magnetic field were clarified, such as the linear growth of dielectric constant with the magnetic field [24], dimensional reduction of the Coulomb field of a point source [25], and the upper bound to the magnetic field due to positronium collapse [26]. Also the notion of anomalous magnetic moment of the photon was introduced [27], especially interesting in the large-field limit owing to the above linear growth [28]. In the meanwhile little attention has been paid to the admixture of electric field. One of the reasons was that although the electric field is generated along the magnetic field lines in the magnetospheres of rotating, strongly magnetized neutron stars [29], the component E$_{‖}$ in the vicinity of all compact astronomical objects mentioned above is sufficiently small ( E$_{‖}$/B ≪ 1) [29] as compared to the magnetic field. Hence, at the first sight, E$_{‖}$ could result in only minor corrections [30]. However, it may be not the case at least for some processes that are forbidden without the electric field ( E$_{‖}$ = 0) and might be allowed at E$_{‖}$ = 0. Splitting of photons in a strong magnetic field ( γ + B → γ ′ + γ ” + B ) is one of the candidates to be such a process. The point is that in a magnetic-like field splitting of one photon mode is allowed, while splitting of the other is strictly forbidden [16, 31]. As far as we are aware, the polarization selection rules for photon splitting have never been 5