Symmetry and uniqueness of the positive solution for the critical Hartree equation on the Heisenberg group

We apply the moving plane method in integral forms to classify the positive solutions of the critical Hartree equation on Heisenberg group \begin{equation}\label{0.1} -\Delta_{\mathbb{H}}u=\left(\int_{\mathbb{H}^{n}}\frac{|u(\xi)|^{Q^{\ast}_{\mu}}}{|\zeta^{-1}\xi|^{\mu}}\mathrm{d}\xi\right)|u|^{Q^{\ast}_{\mu}-2}u,~~~\zeta,\xi\in\mathbb{H}^{n}, \end{equation} where $\Delta_{\mathbb{H}}$ denotes the Kohn Laplacian, $u(\xi)$ is a real-valued function, $Q=2n+2$ is the homogeneous dimension of $\mathbb{H}^{n}$, $\mu\in (0,Q)$ is a real parameter and $Q^{\ast}_{\mu}=\frac{2Q-\mu}{Q-2}$ is the upper critical exponent associated with the Hardy-Littlewood-Sobolev inequality on the Heisenberg group. By introducing the $\mathbb{H}$-reflection, we prove that the solutions of (\ref{0.1}) are cylindrical, upto Heisenberg translation and suitable scaling of function \begin{equation*}\label{0.2} u_{0}(\zeta)=u_{0}(z,t)=\left((1+|z|^{2})^{2}+t^{2}\right)^{-\frac{Q-2}{4}},~~~\zeta=(z,t)\in \mathbb{H}^{n}. \end{equation*} Furthermore, we show that these positive solutions are also CR inversion-symmetric with respect to the unit CC sphere. Consequently, we establish the uniqueness of positive solutions to equation (\ref{0.1}).