  Using $448.0 \times 10^6$~$\psi(3686)$ events collected with the BESIII
detector, an amplitude analysis is performed for
$\psi(3686)\to\gamma\chi_{c1}$, $\chi_{c1}\to\eta\pi^+\pi^-$ decays. The most
dominant two-body structure observed is $a_0(980)^{\pm}\pi^{\mp}$;
$a_0(980)^{\pm}\to\eta\pi^{\pm}$. The $a_0(980)$ line shape is modeled using a
dispersion relation, and a significant non-zero $a_0(980)$ coupling to the
$\eta^{\prime}\pi$ channel is measured. We observe $\chi_{c1}\to a_2(1700)\pi$
production for the first time, with a significance larger than 17$\sigma$. The
production of mesons with exotic quantum numbers, $J^{PC}=1^{-+}$, is
investigated, and upper limits for the branching fractions $\chi_{c1}\to
\pi_1(1400)^{\pm}\pi^{\mp}$, $\chi_{c1}\to \pi_1(1600)^{\pm}\pi^{\mp}$, and
$\chi_{c1}\to\pi_1(2015)^{\pm}\pi^{\mp}$, with subsequent $\pi_1(X)^{\pm} \to
\eta\pi^{\pm}$ decay, are determined.
