Muon decays with lepton--number violation via vector leptoquark

The decays $\mu \rightarrow e \gamma$, $\mu \rightarrow e \gamma \gamma$, and $\mu \rightarrow e e \bar e$ are analysed in the framework of the Pati-Salam type quark-lepton symmetry $SU(4)_V \otimes SU(2)_L \otimes G_R$ where the effects of mixing in the quark-lepton currents are taken into account. It is shown that the $\mu \rightarrow e \gamma \gamma$ and $\mu \rightarrow e e \bar e$ decays via the vector leptoquark have not a GIM--type suppression, while the $\mu \rightarrow e \gamma$ decay has. So, the specific hierarchy of the decay probabilities could take place $\Gamma(\mu \rightarrow e e \bar e) \gg \Gamma(\mu \rightarrow e \gamma \gamma) \gg \Gamma(\mu \rightarrow e \gamma)$. The existing bounds on the vector leptoquark mass and on the mixing matrix elements, based on the data for the $\mu e$ conversion in nuclei and for the ratio of the $K_{e 2}$ and $K_{\mu 2}$ decays allow to set the upper limits on the branching ratios at a level of $10^{-18}$ for the $\mu \rightarrow e \gamma \gamma$ decay and at a level of $10^{-15}$ for the $\mu \rightarrow e e \bar e$ decay.