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