| Summary: | We report a study of the process γγ→X→η[subscript c]π[superscript +]π[superscript -], where X stands for one of the resonances χ[subscript c2](1P), η[subscript c](2S), X(3872), X(3915), or χ[subscript c2](2P). The analysis is performed with a data sample of 473.9 fb-1 collected with the BABAR detector at the PEP-II asymmetric-energy electron-positron collider. We do not observe a significant signal for any channel, and calculate 90% confidence-level upper limits on the products of branching fractions and two-photon widths Γ[subscript ]X→[subscript γγ]B(X→η[subscript c]π[superscript +]π[superscript -]): 15.7 eV for χ[subscript c2](1P), 133 eV for η[subscript c](2S), 11.1 eV for X(3872) (assuming it to be a spin-2 state), 16 eV for X(3915) (assuming it to be a spin-2 state), and 18 eV for χ[subscript c2](2P). We also report upper limits on the ratios of branching fractions B(η[subscript c](2S)→η[subscript c]π[superscript +]π[superscript -])/B(η[subscript c](2S)→K[subscript S][superscript 0]K[superscript +]π[superscript -])<10.0 and B(χ[subscript c2](1P)→η[subscript c]π[superscript +]π[superscript -])/B(χ[subscript c2](1P)→K[subscript S]0K[superscript +]π[superscript -])<32.9 at the 90% confidence level.
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