“…Of interest is also the possibility of establishing a correlation between the scaling law −8/3 observed in the turbulent spectra of the cosmic plasma in the solar wind and magnetosheath (Passot & Sulem 2015;Chefranov & Chefranov 2021) and the similar exponent of the turbulent spectrum of fluctuations in the density of particles of CRs obtained in this work, and also with a similar spectrum of −8/3, characteristic of fluctuations in the intensity of the surface earth electric field (Anisimov et al 2002). It is of interest to further use the obtained exact solutions in the theory of turbulence and in the new theory of Vavilov-Cherenkov radiation (Chefranov 2004(Chefranov , 2012(Chefranov , 2016 to analyze the problems of describing such turbulence and to consider the mechanism of acceleration of CR particles and their origin. In our paper, it is also shown that shock waves can spontaneously form in a medium consisting of CR particles, which, like shock waves arising from the explosion of supernovae, can cause additional particle acceleration.…”