We measure the branching fractions and $CP$ asymmetries for the singly Cabibbo-suppressed decays $D^{0}topi^{+}pi^{-}eta$, $D^{0}to K^{+}K^{-}eta$, and $D^{0}tophieta$, using 980 fb$^{-1}$ of data from the Belle experiment at the KEKB $e^+e^-$ collider. We obtain begin{eqnarray} mathcal{B}(D^{0}topi^{+}pi^{-}eta) & = & [1.22pm 0.02,({rm stat})pm 0.02,({rm syst})pm 0.03,(mathcal{B}_{rm ref})]times 10^{-3},, onumber mathcal{B}(D^{0}to K^{+}K^{-}eta) & = & [1.80,^{+0.07}_{-0.06},({rm stat})pm 0.04,({rm syst})pm 0.05,(mathcal{B}_{rm ref})]times 10^{-4},, onumber mathcal{B}(D^{0}tophieta) & = & [1.84pm 0.09,({rm stat})pm 0.06,({rm syst})pm 0.05,(mathcal{B}_{rm ref})]times 10^{-4},, onumber end{eqnarray} where the third uncertainty ($mathcal{B}_{rm ref}$) is from the uncertainty in the branching fraction of the reference mode $D^{0}to K^{-}pi^{+}eta$. The color-suppressed decay $D^{0}tophieta$ is observed for the first time, with very high significance. The results for the $CP$ asymmetries are begin{eqnarray} A_{CP}(D^{0}topi^{+}pi^{-}eta) & = & [0.9pm 1.2,({rm stat})pm 0.5,({rm syst})]%,, onumber A_{CP}(D^{0}to K^{+}K^{-}eta) & = & [-1.4pm 3.3,({rm stat})pm 1.1,({rm syst})]%,, onumber A_{CP}(D^{0}tophieta)&= & [-1.9pm 4.4,({rm stat})pm 0.6,({rm syst})]%,. onumber end{eqnarray} The results for $D^{0}topi^{+}pi^{-}eta$ are a significant improvement over previous results. The branching fraction and $A_{CP}$ results for $D^{0}to K^{+}K^{-}eta$, and the $A_{CP}$ result for $D^{0}tophieta$, are the first such measurements. No evidence for $CP$ violation is found in any of these decays.
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