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Observation and study of the decay <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>J</mml:mi><mml:mo stretchy="false">/</mml:mo><mml:mi>ψ</mml:mi><mml:mo stretchy="false">→</mml:mo><mml:mi>ϕ</mml:mi><mml:mi>η</mml:mi><mml:msup><mml:mi>η</mml:mi><mml:mo>′</mml:mo></mml:msup></mml:math>
We report the observation and study of the decay $J/\ensuremath{\psi}\ensuremath{\rightarrow}\ensuremath{\phi}\ensuremath{\eta}{\ensuremath{\eta}}^{\ensuremath{'}}$ using $1.3\ifmmode\times\else\texttimes\fi{}1{0}^{9}$ $J/\ensuremath{\psi}$ events collected with the BESIII detector. Its branching fraction, including all possible intermediate states, is measured to be $(2.32\ifmmode\pm\else\textpm\fi{}0.06\ifmmode\pm\else\textpm\fi{}0.16)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}4}$. We also report evidence for a structure, denoted as $X$, in the $\ensuremath{\phi}{\ensuremath{\eta}}^{\ensuremath{'}}$ mass spectrum in the $2.0--2.1\text{ …