$n=1$. So, for simplification, we just give the proof of the equivalence of $\left(
2\right)  \Longleftrightarrow \left(  3\right)  $ for $n=1$.

Because $$d\omega_{1}^{1}=R\theta^{1}\wedge \theta^{\overline{1}}+A_{11,\overline{1}}\theta^{1}\wedge \theta-A_{\overline{1}\overline{1},1}\theta^{\overline{1}}\wedge \theta,$$ we have$$\begin{array}
[c]{ccl}d\left(  \omega_{1}^{1}+iR\theta \right)  & = & d\omega_{1}^{1}+i\left(
R_{,1}\theta^{1}+R_{,\overline{1}}\theta^{\overline{1}}\right)  \wedge
\theta-R\theta^{1}\wedge \theta^{\overline{1}}\\
& = & i\left[  \left(  R_{,1}-iA_{11,\overline{1}}\right)  \theta^{1}+\left(
R_{,\overline{1}}+iA_{\overline{1}\overline{1},1}\right)  \theta^{\overline
{1}}\right]  \wedge \theta
\end{array}
.$$

Hence $$d\left(  \omega_{1}^{1}+iR\theta \right)  =0\Longleftrightarrow R_{,1}-iA_{11,\overline{1}}=0.$$

We recall some useful notations as well.

([@l]) (i) Let $(M,J,\theta)$ be a three-dimensional strictly pseudoconvex CR manifold. We define $$P\varphi=(P_{1}\varphi)\theta^{1},$$ which is an operator that characterizes CR-pluriharmonic functions. Here $P_{1}\varphi=\varphi_{\bar{1}}{}^{\bar{1}}{}_{1}+iA_{11}\varphi^{1}$ and $\overline{P}\varphi=(\overline{P}_{1})\theta^{\bar{1}}$, the conjugate of $P$. The CR Paneitz operator $P_{0}$ is defined by$$P_{0}\varphi=\left(  \delta_{b}(P\varphi)+\overline{\delta}_{b}(\overline
{P}\varphi)\right)  ,\label{id9}$$ where $\delta_{b}$ is the divergence operator that takes $(1,0)$-forms to functions by $\delta_{b}(\sigma_{1}\theta^{1})=\sigma_{1,}{}^{1}$ and, similarly, $\bar{\delta}_{b}(\sigma_{\bar{1}}\theta^{\bar{1}})=\sigma_{\bar
{1},}{}^{\bar{1}}$. We observe that$$\int_{M}\langle P\varphi+\overline{P}\varphi,d_{b}\varphi \rangle_{L_{\theta
}^{\ast}}\ d\mu=-\int_{M}P_{0}\varphi \cdot \varphi \ d\mu \label{10}$$ with $d\mu=\theta \wedge d\theta.$ One can check that $P_{0}$ is self-adjoint, that is, $\left \langle P_{0}\varphi,\psi \right \rangle =\left \langle
\varphi,P_{0}\psi \right \rangle $ for all smooth functions $\varphi$ and $\psi
$. For the details about these operators, the reader can make reference to [@gl], [@h], [@l], [@gg] and [@fh].

\(ii) On a complete pseudohermitian $3$-manifold $(M,J,\theta),$ we call the Paneitz operator $P_{0}$ with 