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

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 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})