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The lepton flavor conserving and lepton flavor violating decays of $K$ and$B$ mesons to lepton pairs in the gauge group $G=SU_L(2)\times U_{Y_1}(1)\times SU_X(2)$ are discussed. The quark-lepton transitions mediated by thelepto-quark bosons $X_{\mu}^{\pm2/3}$ of the group $SU_{X}(2)$ provide aframework to construct an effective Hamiltonian for these decays. The effectivecoupling constant $\frac{G_{X}}{\sqrt{2}}=\frac{g^{2}_{X}}{8m^{2}_{X}}$;$m_{X}$ is the mass at which the group $G$ is broken to the SM group. The upperbound on ($G_{X}/G_{F}$)is obtained from the most stringent experimental limiton the $B.R(K^{0}_{L}\to \mu^{\mp}e^{\pm})$. Several cases of pairing threegenerations of leptons and quarks in the representation $(2,\bar{2})$ of thegroup are analyzed. For some pairing, the upper bound on $(G_{X}/G_{F})$ is ofthe order $(6-9)\times 10^{-6}$ and is compatible with the upper limits onvarious LF violating $K$-decays . In particular for these cases, we find theupper limit on the branching ratio $B.R(K^{0}_{L}\to\mu^{-}\mu^{+})\sim(1.9-8.3)\times10^{-9}$. It is shown that for LF violating B-decays to leptonpairs, the time integrated decay rate$B^{0}_{d,s}\to\ell^{-}\tau^{+}(\tau^{-}\ell^{+})$ is a promising area to testthe model.

Lepton flavor violating decays of mesons to lepton-pairs in a gauge group $SU_L(2)\times U_{Y_1}(1) \times SU_X(2)$