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We report on analyses of tau lepton decays $\tau^- \to \eta K^- \nu_{\tau}$ and $\tau^- \to \eta \pi^- \nu_{\tau}$, with $\eta \to \pi^+ \pi^- \pi^0$, using 470 fb$^{-1}$ of data from the Babar experiment at PEP-II, collected at center-of-mass energies at and near the $\Upsilon(4S)$ resonance. We measure the branching …