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

Using an ${e}^{+}{e}^{-}$ annihilation data sample corresponding to an integrated luminosity of $2.93\text{}\text{}{\mathrm{fb}}^{-1}$ 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 $\left(2.96±0.24±0.10\right)×{10}^{-3}$, $\left(2.23±0.15±\phantom{\rule{0ex}{0ex}}0.10\right)×{10}^{-3}$, and $\left(1.20±0.07±0.04\right)×{10}^{-3}$, respectively, where the first uncertainties are statistical and the second ones are systematic. The ${D}^{+}\to \eta \eta {\pi }^{+}$ decay is observed for the first time, and the branching fractions of ${D}^{+\left(0\right)}\to \eta {\pi }^{+}{\pi }^{0\left(-\right)}$ 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.