Light meson spectroscopy from Dalitz plot analyses of <math><msub><mi>η</mi><mi>c</mi></msub></math> decays to <math><msup><mrow><mi>η</mi></mrow><mrow><mo>′</mo></mrow></msup><msup><mi>K</mi><mo>+</mo></msup><msup><mi>K</mi><mo>−</mo></msup></math>, <math><msup><mrow><mi>η</mi></mrow><mrow><mo>′</mo></mrow></msup><msup><mi>π</mi><mo>+</mo></msup><msup><mi>π</mi><mo>−</mo></msup></math>, and <math><mi>η</mi><msup><mi>π</mi><mo>+</mo></msup><msup><mi>π</mi><mo>−</mo></msup></math> produced in two-photon interactions

We study the processes γγ→ηc→η′K+K−, η′π+π−, and ηπ+π− 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. This is the first observation of the decay ηc→η′K+K− and we measure the branching fraction Γ(ηc→η′K+K−)/(Γ(ηc→η′π+π−)=0.644&#177;0.039stat&#177;0.032sys. Significant interference is observed between γγ→ηc→ηπ+π− and the nonresonant two-photon process γγ→ηπ+π−. A Dalitz plot analysis is performed of ηc decays to η′K+K−, η′π+π−, and ηπ+π−. Combined with our previous analysis of ηc→KK&#175;π, we measure the K0*(1430) parameters and the ratio between its η′K and πK couplings. The decay ηc→η′π+π− is dominated by the f0(2100) resonance, also observed in J/ψ radiative decays. A new a0(1700)→ηπ resonance is observed in the ηc→ηπ+π− channel. We also compare ηc decays to η and η′ final states in association with scalar mesons as they relate to the identification of the scalar glueball.