  In this paper we prove symmetry of nonnegative solutions of the integral
equation \[ u (\zeta ) = \int\limits_{{\mathbb H}^n} |\zeta^{-1}
\xi|^{-(Q-\alpha)} u(\xi)^{p} d\xi \quad 1< p \leq
\frac{Q+\alpha}{Q-\alpha},\quad 0< \alpha <Q \] on the Heisenberg group
${\mathbb H}^n = {\mathbb C}^n \times {\mathbb R}$, $Q= 2n +2$ using the moving
plane method and the Hardy-Littlewood-Sobolev inequality proved by Frank and
Lieb for the Heisenberg group. For $p$ subcritical, i.e., $1< p <
\frac{Q+\alpha}{Q-\alpha}$ we show nonexistence of positive solution of this
integral equation, while for the critical case, $p = \frac{Q+\alpha}{Q-\alpha}$
we prove that the solutions are cylindrical and are unique upto Heisenberg
translation and suitable scaling of the function \[ u_0 (z,t) = \left( (1+
|z|^2)^2 + t^2 \right)^{- \frac{Q-\alpha}{4}} \quad (z,t ) \in {\mathbb H}^n.
\] As a consequence, we also obtain the symmetry and classification of
nonnegative $C^2$ solution of the equation \[ \Delta_{\mathbb H} u + u^{p} = 0
\quad \mbox{for } 1< p \leq \frac{Q+\alpha}{Q-\alpha} \mbox{ in } {\mathbb H}^n
\] without any partial symmetry assumption on the function $u$.
