  We calculate the coupling constants of the decay modes
$1^{-+}\rightarrow\rho\pi, f_1\pi, b_1\pi, \eta\pi, \eta'\pi, a_1\pi, f_1\eta$
within the framework of the light-cone QCD sum rule. Then we calculate the
partial width of these decay channels, which differ greatly from the existing
calculations using phenomenological models. For the isovector $1^{-+}$ state,
the dominant decay modes are $\rho\pi, f_1\pi$. For its isoscalar partner, its
dominant decay mode is $a_1\pi$. We also discuss the possible search of the
$1^{-+}$ state at BESIII, for example through the decay chains $J/\psi
(\psi')\to \pi_1 +\gamma$ or $J/\psi (\psi')\to \pi_1 +\rho$ where $\pi_1$ can
be reconstructed through the decay modes $\pi_1\to \rho\pi\to \pi^+\pi^-\pi^0$
or $\pi_1\to f_1(1285)\pi^0$. Hopefully the present work will be helpful to the
experimental establishment of the $1^{-+}$ hybrid meson.

High Energy Physics - Phenomenology;High Energy Physics - Experiment;Nuclear Experiment;Nuclear Theory