Studies of<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:msup><mml:mi>τ</mml:mi><mml:mo>−</mml:mo></mml:msup><mml:mo>→</mml:mo><mml:mi>η</mml:mi><mml:msup><mml:mi>K</mml:mi><mml:mo>−</mml:mo></mml:msup><mml:msub><mml:mi>ν</mml:mi><mml:mi>τ</mml:mi></mml:msub></mml:math>and<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:msup><mml:mi>τ</mml:mi><mml:mo>−</mml:mo></mml:msup><mml:mo>→</mml:mo><mml:mi>η</mml:mi><mml:msup><mml:mi>π</mml:mi><mml:mo>−</mml:mo></mml:msup><mml:msub><mml:mi>ν</mml:mi><mml:mi>τ</mml:mi></mml:msub></mml:math>at<i>BABAR</i>and a search for a second-class current

We report on analyses of tau lepton decays ${\ensuremath{\tau}}^{\ensuremath{-}}\ensuremath{\rightarrow}\ensuremath{\eta}{K}^{\ensuremath{-}}{\ensuremath{\nu}}_{\ensuremath{\tau}}$ and ${\ensuremath{\tau}}^{\ensuremath{-}}\ensuremath{\rightarrow}\ensuremath{\eta}{\ensuremath{\pi}}^{\ensuremath{-}}{\ensuremath{\nu}}_{\ensuremath{\tau}}$, with $\ensuremath{\eta}\ensuremath{\rightarrow}{\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{\ensuremath{-}}{\ensuremath{\pi}}^{0}$, using $470\text{ }\text{ }{\mathrm{fb}}^{\ensuremath{-}1}$ of data from the BABAR experiment at PEP-II, collected at center-of-mass energies at and near the $\mathit{\ensuremath{\Upsilon}}(4S)$ resonance. We measure the branching fraction for the ${\ensuremath{\tau}}^{\ensuremath{-}}\ensuremath{\rightarrow}\ensuremath{\eta}{K}^{\ensuremath{-}}{\ensuremath{\nu}}_{\ensuremath{\tau}}$ decay mode, $\mathcal{B}({\ensuremath{\tau}}^{\ensuremath{-}}\ensuremath{\rightarrow}\ensuremath{\eta}{K}^{\ensuremath{-}}{\ensuremath{\nu}}_{\ensuremath{\tau}})=(1.42\ifmmode\pm\else\textpm\fi{}0.11(\mathrm{stat})\ifmmode\pm\else\textpm\fi{}0.07(\mathrm{syst}))\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}4}$, and report a 95% confidence level upper limit for the second-class current process ${\ensuremath{\tau}}^{\ensuremath{-}}\ensuremath{\rightarrow}\ensuremath{\eta}{\ensuremath{\pi}}^{\ensuremath{-}}{\ensuremath{\nu}}_{\ensuremath{\tau}}$, $\mathcal{B}({\ensuremath{\tau}}^{\ensuremath{-}}\ensuremath{\rightarrow}\ensuremath{\eta}{\ensuremath{\pi}}^{\ensuremath{-}}{\ensuremath{\nu}}_{\ensuremath{\tau}})&lt;9.9\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}5}$.