  With a data sample corresponding to an integrated luminosity of
11.5~fb$^{-1}$
  collected with the BESIII detector operating at the BEPCII storage ring, for
the first time the light hadron decay $\chi_{c1}(3872) \rightarrow
\pi^{+}\pi^{-}\eta$
  is searched for. While no significant signal is observed, the upper limits at
the 90\% confidence level for
  $\sigma[e^{+}e^{-} \rightarrow \gamma \chi_{c1}(3872)]
\mathcal{B}[\chi_{c1}(3872) \rightarrow \pi^{+}\pi^{-}\eta]$ at center-of-mass
energies from 4.13 to 4.34 GeV are determined.
  By normalizing to the $\chi_{c1}(3872)\to\pi^+\pi^- J/\psi$ decay channel, a
90\% confidence level upper limit for the branching fraction ratio
  $\mathcal{R}=\mathcal{B}[\chi_{c1}(3872)
\rightarrow\pi^{+}\pi^{-}\eta]/\mathcal{B}[\chi_{c1}(3872) \rightarrow
\pi^{+}\pi^{-} J/\psi] < 0.12$ is given.
  These measurements provide important inputs for understanding the internal
structure of the $\chi_{c1}(3872)$ resonance.
