Search for $psi(3686)to gammaeta_c(eta(1405))to gamma pi^{+}pi^{-}pi^{0}$

Using a sample of $448.1times10^{6}$ $psi(3686)$ events collected with the BESIII detector, a search for the isospin violating decay $eta_{c}topi^{+}pi^{-}pi^{0}$ via $psi(3686)togammaeta_{c}$ is presented. No signal is observed, and the upper limit on $mathcal{B}(psi(3686)to gammaeta_{c}to gammapi^{+}pi^{-}pi^{0} )$ is determined to be $1.6times10^{-6}$ at the $90%$ confidence level. In addition, a search for $eta(1405)to f_{0}(980)pi^{0}$ in $psi(3686)$ radiative decays is performed. No signal is observed, and the branching fraction $mathcal{B}(psi(3686)togammaeta(1405)to gamma f_{0}(980)pi^{0}togamma pi^+pi^-pi^0 )$ is calculated to be less than $ 5.0times10^{-7}$ at the $90%$ confidence level.