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We present the preliminary results from the search of the non-conservation of
lepton flavor number in the decay of a $\tau$ to a lighter mass lepton
and a pseudo-scalar meson, performed using $e^+e^- \rightarrow \tau^+\tau^-$ events collected
at a center-of-mass energy near 10.58 GeV with the BABAR detector at the SLAC PEP-II $e^+e^-$ storage ring.
No evidence of such a signal has been found in the data sample corresponding to a luminosity of 314.5 fb$^{-1}$,
and we set an upper limit of $1.6\times 10^{-7}$ at 90\% confidence level on the decay of $\tau^- \rightarrow \mu^-\eta$.
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