Search for $X\left(1870\right)$ via the decay $J/\psi \to \omega {K}^{+}{K}^{-}\eta $

Using a sample of $\left(10087±44\right)×{10}^{6}\text{}\text{}J/\psi $ events collected by the BESIII detector at the BEPCII collider, we search for the decay $X\left(1870\right)\to {K}^{+}{K}^{-}\eta $ via the $J/\psi \to \omega {K}^{+}{K}^{-}\eta $ process for the first time. No significant $X\left(1870\right)$ signal is observed. The upper limit on the branching fraction of the decay $J/\psi \to \omega X\left(1870\right)\to \omega {K}^{+}{K}^{-}\eta $ is determined to be $9.55×{10}^{-7}$ at the 90% confidence level. In addition, the branching faction $B\left(J/\psi \to \omega {K}^{+}{K}^{-}\eta \right)$ is measured to be $\left(3.33±0.02\left(\mathrm{stat}\right)±0.12\left(\mathrm{syst}\right)\right)×{10}^{-4}$.