Search for 1^-+ Charmoniumlike Hybrid Via <i>e</i>⁺<i>e</i>^- → Γη⁽′⁾η<i>_c</i> at Center-of-mass Energies Between 4.258 and 4.681 Gev

Using e(+)e(-) collision data corresponding to an integrated luminosity of 10.6 fb(-1) collected at center-of-mass energies between 4.258 and 4.681 GeV with the BESIII detector at the BEPCII collider, we search for the 1(-+) charmoniumlike hybrid via e(+)e(-) -> gamma eta eta(c) and e(+)e(-) -> gamma eta 'eta(c)decays for the first time. No significant signal is observed, and the upper limits on the Born cross sections for both processes are set at the 90% confidence level.