$Z$ boson decays $Z\rightarrow{{l_i}^{\pm}{l_j}^{\mp}}$ and Higgs boson decays $h\rightarrow{{l_i}^{\pm}{l_j}^{\mp}}$ with lepton flavor violation in a $U(1)$ extension of the MSSM
$U(1)_X$SSM is the extension of the minimal supersymmetric standard model (MSSM) and its local gauge group is $SU(3)_C\times SU(2)_L \times U(1)_Y \times U(1)_X$. We study lepton flavor violating (LFV) decays $Z\rightarrow{{l_i}^{\pm}{l_j}^{\mp}}$($Z\rightarrow eμ$, $Z\rightarrow eτ$, and $Z\rightarrow μτ$) and $h\rightarrow{{l_i}^{\pm}{l_j}^{\mp}}$($h\rightarrow eμ$, $h\rightarrow eτ$, and $h\rightarrow μτ$), in this model. In the numerical results, the branching ratios of $Z\rightarrow{{l_i}^{\pm}{l_j}^{\mp}}$ are from $10^{-9}$ to $10^{-13}$ and the branching ratios of $h\rightarrow{{l_i}^{\pm}{l_j}^{\mp}}$ are from $10^{-3}$ to $10^{-9}$, which can approach the present experimental upper bounds. Based on the latest experimental data, we analyze the influence of different sensitive parameters on the branching ratio, and make reasonable predictions for future experiments. The main sensitive parameters and LFV sources are the non-diagonal elements corresponding to the initial and final generations of leptons, which can be seen from the numerical analysis.