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Search for intermediate resonances and dark gauge bosons in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:mi>J</mml:mi><mml:mo stretchy="false">/</mml:mo><mml:mi>ψ</mml:mi><mml:mo stretchy="false">→</mml:mo><mml:mi>γ</mml:mi><mml:msup><mml:mrow><mml:mi>π</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msup><mml:msup><mml:mrow><mml:mi>η</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo></mml:mrow></mml:msup></mml:mrow></mml:math>

We report on an analysis of the decay $J/\ensuremath{\psi}\ensuremath{\rightarrow}\ensuremath{\gamma}{\ensuremath{\pi}}^{0}{\ensuremath{\eta}}^{\ensuremath{'}}$ using a sample of $(1310.6\ifmmode\pm\else\textpm\fi{}7.0)\ifmmode\times\else\texttimes\fi{}{10}^{6}J/\ensuremath{\psi}$ events collected with the BESIII detector. We search for the $CP$-violating process ${\ensuremath{\eta}}_{c}\ensuremath{\rightarrow}{\ensuremath{\pi}}^{0}{\ensuremath{\eta}}^{\ensuremath{'}}$ and a dark gauge boson ${U}^{\ensuremath{'}}$ in $J/\ensuremath{\psi}\ensuremath{\rightarrow}{U}^{\ensuremath{'}}{\ensuremath{\eta}}^{\ensuremath{'}},{U}^{\ensuremath{'}}\ensuremath{\rightarrow}\ensuremath{\gamma}{\ensuremath{\pi}}^{0},{\ensuremath{\pi}}^{0}\ensuremath{\rightarrow}\ensuremath{\gamma}\ensuremath{\gamma}$. No evidence of an ${\ensuremath{\eta}}_{c}$ signal is observed in the ${\ensuremath{\pi}}^{0}{\ensuremath{\eta}}^{\ensuremath{'}}$ invariant-mass spectrum and the …