Search for c (2S) 2 (^+^-) and improved measurement of ₂₉ 2 (^+^-)

We search for the hadronic decay c (2S) 2 (^+^-) in the (3686) c (2S) radiative decay using (27. 12 0. 14) 10⁸ (3686) events collected by the BESIII detector at the BEPCII collider. No significant signal is found, and the upper limit of B (3686) c (2S) Bc (2S) 2 (^+^-) is determined to be 0. 78 10^-6 at the 90\% confidence level. Using (3686) ₂₉ transitions, we also measure the branching fractions of B₂₉ (₉=₀, ₁, ₂) 2 (^+^-), which are B₂₀ 2 (^+^-) = (2. 127 0. 002~ (stat. ) 0. 101~ (syst. ) ) \%, B₂₁ 2 (^+^-) = (0. 685 0. 001~ (stat. ) 0. 031~syst. ) ) \%, and B₂₂ 2 (^+^-) = (1. 153 0. 001~ (stat. ) 0. 063~ (syst. ) ) \%.