We report on analyses of tau lepton decays $tau^- to eta K^- u_{tau}$ and $tau^- to eta pi^- u_{tau}$, 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 fraction for the $tau^- to eta K^- u_{tau}$ decay mode, $Br(tau^- to eta K^- u_{tau}) = (1.42pm0.11text{(stat)}pm0.07text{(syst)})times10^{-4}$, and report a 95% confidence level upper limit for the second-class current process $tau^- to eta pi^- u_{tau}$, $Br(tau^- to eta pi^- u_{tau}) < 9.9times10^{-5}$.
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