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Considering the constraints of the lepton flavor violating (LFV) processes$\mu \rightarrow 3e$ and $\tau\rightarrow3\mu$ on the LFV couplings$Z'\ell_{i}\ell_{j}$, in the contexts of the $E_{6}$ models, the left-right(LR) models, the "alternative" left-right (ALR) models and the 331 models, weinvestigate the contributions of the extra gauge boson $Z'$ to the decay ratesof the processes $\ell_{i}\rightarrow\ell_{j}\nu_{\ell}\nu_{\ell}$,$\tau\rightarrow\mu P $ and $P\rightarrow \mu e$ with $P=\pi^{0},\eta$ and$\eta '$. Our numerical results show that the maximal values of the branchingratios for these processes are not dependent on the $Z'$ mass $M_{Z'}$ atleader order. The extra gauge boson $Z'_{X}$ predicted by the $E_{6}$ modelscan make the maximum value of the branching ratio$Br(\tau\rightarrow\mu\nu_{\ell}\nu_{\ell})$ reach $1.1\times10^{-7}$. All $Z'$models considered in this paper can produce significant contributions to theprocess $\tau\rightarrow\mu P$. However, the value of $Br(P\rightarrow\mu e)$is far below its corresponding experimental upper bound.

LFV couplings of the extra gauge boson Z' and leptonic decay and production of pseudoscalar mesons