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Recently, the Belle collaboration reported the first measurements of the$\tau^- \to\nu_{\tau}\pi^-e^+e^-$ branching fraction and the spectrum of thepion-dielectron system. In an analysis previous to Belle's results, weevaluated this branching fraction which turned out to be compatible with thatreported by Belle, although with a large uncertainty. This is the motivation toseek for improvements on our previous evaluation of $\tau^- \to\nu_{\tau}\pi^-\ell^+\ell^-$ decays ($\ell=e,\,\mu$). In this paper we improveour calculation of the $WP^-\gamma^*$ vertex by including flavour symmetrybreaking effects in the framework of the Resonance Chiral Theory. We impose QCDshort-distance behaviour to constrain most parameters and data on the$\pi^-e^+e^-$ spectrum reported by Belle to fix the remaining free ones. As aresult, improved predictions for the branching ratios and hadronic/leptonicspectra are reported, in good agreement with observations. Analogouscalculations for the strangeness-changing $\tau^- \to\nu_{\tau}K^-\ell^+\ell^-$ transitions are reported for the first time. Albeitone expects the $m_{\pi\mu^+\mu^-}$ spectrum to be measured in Belle-II and theobservables with $\ell=e$ can be improved, it is rather unlikely that the $K$channels can be measured due to the suppression factor$|V_{ud}/V_{us}|^2=0.05$.

Improved description of di-lepton production in $\tau^-\to\nu_\tau P^-$ decays