Amplitude analysis of $D_{s}^{+} rightarrow pi^{+}pi^{0}eta$ and first observation of the pure $W$-annihilation decays $D_{s}^{+} rightarrow a_{0}(980)^{+}pi^{0}$ and $D_{s}^{+} rightarrow a_{0}(980)^{0}pi^{+}$

We present the first amplitude analysis of the decay $D^{+}_{s} rightarrow pi^{+}pi^{0}eta$. We use an $e^{+}e^{-}$ collision data sample corresponding to an integrated luminosity of 3.19~${mbox{,fb}^{-1}}$ collected with the BESIII detector at a center-of-mass energy of $4.178$ GeV. We observe for the first time the pure $W$-annihilation decays $D_{s}^{+} rightarrow a_{0}(980)^{+}pi^{0}$ and $D_{s}^{+} rightarrow a_{0}(980)^{0}pi^{+}$. We measure the absolute branching fractions $mathcal{B}(D_{s}^{+} rightarrow a_{0}(980)^{+(0)}pi^{0^(+)}, a_{0}(980)^{+(0)} to pi^{+(0)}eta) = (1.46pm0.15_{{rm stat.}}pm0.23_{{rm sys.}})$%, which is larger than the branching fractions of other measured pure $W$-annihilation decays by at least one order of magnitude. In addition, we measure the branching fraction of $D_{s}^{+} rightarrow pi^{+}pi^{0}eta$ with significantly improved precision.