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From analysis of data produced by the BABAR experiment, the first upperbounds (90% C.L.) were obtained on the branching ratios $Br(\tau \to e\gamma\gamma)<2.5 \times 10^{-4}$ and $Br(\tau \to \mu \gamma\gamma)<5.8 \times10^{-4}$. In addition, improved upper bounds (95% C.L.) were found on branchingratios $Br(\tau \to e X)<1.4 \times 10^{-3}$ and $Br(\tau \to \mu X) <2.0\times 10^{-3}$, where $X$ is an undetected weakly interacting boson with mass$m_X<1.6$ GeV/$c^2$.

Upper limits on branching ratios of the lepton-flavor-violating decays $\tau \to \ell \gamma\gamma$ and $\tau \to \ell X$