>10^{17}$ eV). The universal character of the lateral distribution of the energy deposit in the atmosphere, from which the lateral distribution of the fluorescence light emission can be straightforwardly obtained [@gora], is a consequence of the universality of the lateral distributions of electrons with fixed energies. Nerling et al. (2006) and Giller $\&$ Wieczorek (2009) applied the universal angular distributions for deriving the production of the Cherenkov light in showers, although in an approximate way.\
The work described so far concerned distributions of one variable, electron angle or its lateral distance, integrated over the second one. However, it does not present an exact description of the full electron state in a shower since the two distributions are not independent. The electron angular distribution depends on their lateral distance. The first attempt to allow for this was made by Giller et al. (2015) where the dependence of the angular distribution on the lateral distance was considered. However, the distribution of polar angles was assumed to be independent of the azimuth (with respect to the shower axis) what was not quite true (see the present paper). Nevertheless, it was pointed out that thanks to the universal character of all electron distributions a 