Charged pion production in the annihilation reactions $e^+ + e^- \to p + \bar{n} +\pi^-$ and $e^+ + e^- \to n + \bar{p} +\pi^+$

The nonresonant mechanism in the reactions $e^+\,e^-\to \pi^-\,p\,\bar{n}$ and $e^+\,e^-\to \pi^+\,n\,\bar{p}$ is investigated in frame of the one-photon exchange approximation. The description of the hadronic phase space and the invariant amplitude formalism, earlier developed for the neutral pion channel, are applied here. Some modifications are required to fulfill the invariance and to account for an additional contribution due to diagram with the pion pole. Two different variants which lead to dipole and monopole asymptotic behaviour of the pion electromagnetic form factor, are considered. The double and single distributions over the invariant variables as well the total cross section, are obtained. The results are plotted for $s=$5,\,6,\,10,\,16\, GeV$^2$. The calculations are performed for the $\pi^-\,p\,\bar{n}$ channel and the rules to proceed to the $\pi^+\,n\,\bar{p}$ channel are formulated for every distribution.