  Lepton Flavor Violating (LFV) decays are forbidden in the Standard Model (SM)
and to explore them one has to go beyond it. The flavor changing neutral
current induced Lepton Flavor Conserving and LFV decays of $K$ and $B$ mesons
are discussed in the gauge group $G = SU(2)_L\times U(1)_{Y_1}\times SU(2)_X$.
The lepto-quark $X^{\pm 2/3}_{\mu}$ corresponding to gauge group $ SU(2)_X$
allows the quark-lepton transitions and hence giving a framework to construct
the effective Lagrangian for the LFV decays. The mass of lepto-quark $m_{X}$
provides a scale at which the gauge group $G$ is broken to the SM gauge group.
Using the most stringent experimental limit $\mathcal{B}(K^{0}_{L}\to
\mu^{\mp}e^{\pm}) < 1.7 \times 10^{-12}$, the upper bound on the effective
coupling constant $(\frac{G_{X}}{G_F})^2 < 1.1 \times 10^{-10}$ is obtained for
certain pairing of lepton and quark generations in the representation $(2,
\bar{2})$ of the group $G$. Later, the effective Lagrangian for the LFV meson
decays for the gauge group $G = \left[SU(2)_{L}\times SU(2)_{R}\times
U(1)_{Y_1^{\prime}}\right]\times SU(2)_{X}$ is constructed. Using
$\mathcal{B}(K^{-} \to \pi^{-}\nu \bar{\nu}) = (1.7 \pm 1.1)\times 10^{-10}$,
the bound on the ratio of effective couplings is obtained to be
$\left(\frac{G_{X}}{G_{F}}\right)^2 < 10^{-10}$. A number of decay modes are
discussed which provide a promising area to test this model in the current and
future particle physics experiments.
