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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 atcenter-of-mass energies at and near the $\Upsilon(4S)$ resonance. We measurethe branching fraction for the $\tau^- \to \eta K^- \nu_{\tau}$ decay mode,$\Br(\tau^- \to \eta K^- \nu_{\tau}) =(1.42\pm0.11\text{(stat)}\pm0.07\text{(syst)})\times10^{-4}$, and report a 95%confidence level upper limit for the second-class current process $\tau^- \to\eta \pi^- \nu_{\tau}$, $\Br(\tau^- \to \eta \pi^- \nu_{\tau}) <9.9\times10^{-5}$.

Studies of tau- -> eta.K-.nu and tau- -> eta.pi-.nu at BaBar and a search for a second-class current