Observation of $D^{+}\to \eta \eta {\pi }^{+}$ and improved measurement of $D^{0(+)}\to \eta {\pi }^{+}{\pi }^{-(0)}$

Using an $e^{+}{e}^{-}$ annihilation data sample corresponding to an integrated luminosity of $2.93\mathrm{f}{\mathrm{b}}^{-1}$ collected at the center-of-mass energy of 3.773,GeV with the BESIII detector, we measure the absolute branching fractions of $D^{+}\to \eta \eta {\pi }^{+}$, $D^{+}\to \eta {\pi }^{+}{\pi }^0$, and $D^0\to \eta {\pi }^{+}{\pi }^{-}$ to be $(2.96\pm 0.24\pm 0.13)\times {10}^{-3}$, $(2.23 \pm 0.15 \pm 0.11)\times 10^{-3}$,and$(1.20 \pm 0.07 \pm 0.04)\times 10^{-3}$, respectively, where the first uncertainties are statistical and the second ones systematic. The $D^{+}\to \eta \eta {\pi }^{+}$ decay is observed for the first time and the branching fractions of $D^{+(0)}\to \eta {\pi }^{+}{\pi }^{0(-)}$ are measured with much improved precision. In addition we test for $CP$ asymmetries in the separated charge-conjugate branching fractions; no evidence of $CP$ violation is found.