  We study $e^{+}e^{-}$ collisions with a $\pi^{+}\pi^{-}\pi^{0}\eta_{c}$ final
state using data samples collected with the BESIII detector at center-of-mass
energies $\sqrt{s}=4.226$, $4.258$, $4.358$, $4.416$, and $4.600$ GeV. Evidence
for the decay $\zcpm\to\rhopm\etac$ is reported with a statistical significance
of $3.9\sigma$ with various systematic uncertainties taken into account at
$\sqrt{s} = 4.226$ GeV, and the Born cross section times branching fraction
$\sigma^{B}(\EE\to \pimp\zcpm)\times \BR(\zcpm\to\rhopm\etac)$ is measured to
be $(48 \pm 11 \pm 11)\,\rm{pb}$. The $\zcpm\to \rhopm\etac$ signal is not
significant at the other center-of-mass energies and the corresponding upper
limits are determined. In addition, no significant signal is observed in a
search for $\zcppm\to \rho^{\pm}\etac$ with the same data samples. The ratios
$R_{\zc}=\BR(\zcpm\to \rho^{\pm} \etac)/\BR(\zcpm\to \pi^{\pm} \jpsi)$ and
$R_{\zcp}=\BR(\zcppm\to \rho^{\pm} \etac)/\BR(\zcppm\to \pi^{\pm} \hc)$ are
obtained and used to discriminate between different theoretical interpretations
of the $\zcpm$ and $\zcppm$.
