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We search for ${B}^{0}$ meson decays into two-body combinations of ${K}^{0}$, $\ensuremath{\eta}$, ${\ensuremath{\eta}}^{\ensuremath{'}}$, and $\ensuremath{\phi}$ mesons in $324\ifmmode\times\else\texttimes\fi{}{10}^{6}$ $B\overline{B}$ pairs collected with the BABAR detector at the PEP-II asymmetric-energy ${e}^{+}{e}^{\ensuremath{-}}$ collider at SLAC. We measure the following branching fractions (upper limits at 90% confidence level) in units of ${10}^{\ensuremath{-}6}$: $\mathcal{B}({B}^{0}\ensuremath{\rightarrow}\ensuremath{\eta}{K}^{0})={1.8}_{\ensuremath{-}0.6}^{+0.7}\ifmmode\pm\else\textpm\fi{}0.1(<2.9)$, $\mathcal{B}({B}^{0}\ensuremath{\rightarrow}\ensuremath{\eta}\ensuremath{\eta})={1.1}_{\ensuremath{-}0.4}^{+0.5}\ifmmode\pm\else\textpm\fi{}0.1(<1.8)$, $\mathcal{B}({B}^{0}\ensuremath{\rightarrow}\ensuremath{\eta}\ensuremath{\phi})=0.1\ifmmode\pm\else\textpm\fi{}0.2\ifmmode\pm\else\textpm\fi{}0.1(<0.6)$, $\mathcal{B}({B}^{0}\ensuremath{\rightarrow}{\ensuremath{\eta}}^{\ensuremath{'}}\ensuremath{\phi})={0.2}_{\ensuremath{-}0.3}^{+0.4}\ifmmode\pm\else\textpm\fi{}0.1(<1.0)$, and $\mathcal{B}({B}^{0}\ensuremath{\rightarrow}{\ensuremath{\eta}}^{\ensuremath{'}}{\ensuremath{\eta}}^{\ensuremath{'}})={1.0}_{\ensuremath{-}0.6}^{+0.8}\ifmmode\pm\else\textpm\fi{}0.1(<2.4)$, where the first error is statistical and the second systematic.

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