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The process of electron-positron annihilation into $\ensuremath{\eta}({\ensuremath{\eta}}^{\ensuremath{'}})2\ensuremath{\pi}$ is described within the extended Nambu--Jona-Lasinio model in the energy range up to about 2 GeV. Contributions of intermediate vector mesons $\ensuremath{\rho}(770)$ and $\ensuremath{\rho}(1450)$ are taken into account. Results for the $\ensuremath{\eta}2\ensuremath{\pi}$ channel are found to be in a reasonable agreement with experimental data. Predictions for production of ${\ensuremath{\eta}}^{\ensuremath{'}}2\ensuremath{\pi}$ are given. The corresponding estimations for decays $\ensuremath{\tau}\ensuremath{\rightarrow}\ensuremath{\eta}({\ensuremath{\eta}}^{\ensuremath{'}})2\ensuremath{\pi}\ensuremath{\nu}$ are given in the Appendix.

In the extended Nambu--Jona-Lasinio (NJL) model the total cross sections of the processes ${e}^{+}{e}^{\ensuremath{-}}\ensuremath{\rightarrow}[\ensuremath{\eta},{\ensuremath{\eta}}^{\ensuremath{'}},\ensuremath{\eta}(1295),\ensuremath{\eta}(1405)]\ensuremath{\gamma}$ at energies up to 2 GeV are calculated. The intermediate vector mesons $\ensuremath{\rho}(770)$, $\ensuremath{\omega}(782)$, $\ensuremath{\phi}(1020)$, ${\ensuremath{\rho}}^{\ensuremath{'}}(1450)$, ${\ensuremath{\omega}}^{\ensuremath{'}}(1420),$ and ${\ensuremath{\phi}}^{\ensuremath{'}}(1680)$ are taken into account. The latter three mesons are treated as the first radial excited states. They are incorporated into the NJL model by means of polynomial form factors. The calculation results are in satisfactory agreement with the experimental data obtained with the SND detector at VEPP-2M, Novosibirsk. The predictions given for the process ${e}^{+}{e}^{\ensuremath{-}}\ensuremath{\rightarrow}[{\ensuremath{\eta}}^{\ensuremath{'}},\ensuremath{\eta}(1295),\ensuremath{\eta}(1405)]\ensuremath{\gamma}$ have not been experimentally tested yet.

The width of the decay $\ensuremath{\tau}\ensuremath{\rightarrow}\ensuremath{\pi}\ensuremath{\omega}\ensuremath{\nu}$ is calculated in the framework of an extended Nambu-Jona-Lasinio model. The contributions of intermediate $\ensuremath{\rho}(770)$ and ${\ensuremath{\rho}}^{\ensuremath{'}}(1450)$ vector mesons are taken into account. The computed partial width and spectral function of this decay are found to be in a satisfactory agreement with experimental data.

The widths of the decays ${\ensuremath{\rho}}^{\ensuremath{-}}\ensuremath{\rightarrow}\ensuremath{\eta}{\ensuremath{\pi}}^{\ensuremath{-}}$ and ${\ensuremath{\tau}}^{\ensuremath{-}}\ensuremath{\rightarrow}\ensuremath{\eta}({\ensuremath{\eta}}^{\ensuremath{'}}){\ensuremath{\pi}}^{\ensuremath{-}}\ensuremath{\nu}$ are calculated in the framework of the Nambu--Jona-Lasinio model. It is shown that these decays are defined by the $u$ and $d$ quark mass difference. It leads to the suppression of these decays in comparison with the main decay modes. In the process ${\ensuremath{\rho}}^{\ensuremath{-}}\ensuremath{\rightarrow}\ensuremath{\eta}{\ensuremath{\pi}}^{\ensuremath{-}}$ the intermediate scalar ${a}_{0}^{\ensuremath{-}}$ state is taken into account. For the $\ensuremath{\tau}$ decays the intermediate states with ${a}_{0}^{\ensuremath{-}}$, ${\ensuremath{\rho}}^{\ensuremath{-}}(770)$ and ${\ensuremath{\rho}}^{\ensuremath{-}}(1450)$ mesons are used. Our estimates are compared with the results obtained in other works.