We study the processes γγ→K+K−η and γγ→K+K−π0 using a data sample of 519 fb−1 recorded with the BABAR detector operating at the SLAC PEP-II asymmetric-energy e+e− collider at center-of-mass energies at and near the ϒ(nS) (n=2,3,4) resonances. We observe ηc→K+K−π0 and ηc→K+K−η decays, measure their relative branching fraction, and perform a Dalitz plot analysis for each decay. We observe the K0*(1430)→Kη decay and measure its branching fraction relative to the Kπ decay mode to be R(K0*(1430))=B(K0*(1430)→Kη)B(K0*(1430)→Kπ)=0.092±0.025−0.025+0.010. The ηc→K+K−η and K0*(1430)→Kη results correspond to the first observations of these channels. The data also show evidence for ηc(2S)→K+K−π0 and first evidence for ηc(2S)→K+K−η.6 MoreReceived 27 March 2014DOI:https://doi.org/10.1103/PhysRevD.89.112004© 2014 American Physical Society