\theta \overline{^{1}}.
\end{array}
.$$

Due to $$d\sigma=d\omega_{1}^{1}=R_{1\overline{1}}\theta^{1}\wedge \theta^{\overline{1}}+A_{11,\overline{1}}\theta^{1}\wedge \theta-A_{\overline{1}\overline{1},1}\theta^{\overline{1}}\wedge \theta,$$ we derive$$\left \{
\begin{array}
[c]{l}R=R_{1\overline{1}}=\sigma_{\overline{1},1}+\sigma_{1,\overline{1}}-\sigma
_{0}\\
A_{11,\overline{1}}=\sigma_{1,0}+i\sigma_{0,1}-A_{11}\sigma_{\overline{1}}\end{array}
\right.  .$$

We would need the J.J. Kohn’s Hodge theory for the $\overline{\partial}_{b}$ complex (see [@k]) :

\[l3\]If $(M,J,\theta)$ is an embeddable closed strictly pseudoconvex CR $3$-manifold and $\eta \in \Omega^{0,1}\left(  M\right)  $, a smooth $\left(
0,1\right)  $-form on $M$ with $$\overline{\partial}_{b}\eta=0,$$ then there are a complex function $\varphi \in C_{\mathbb{C}
}^{\infty}\left(  M\right)  $ and $\gamma \in \Omega^{0,1}\left(  M\right)  $ such that $$\left(  \eta-\overline{\partial}_{b}\varphi \right)  =\gamma \in \ker \left(
\square_{b}\right)  ,$$ where $\square_{b}=2\left(  \overline{\partial}_{b}\overline{\partial}_{b}^{\ast}+\overline{\partial}_{b}^{\ast}\overline{\partial}_{b}\right)  $ is the Kohn-Rossi Laplacian.

Subsequently, we deduce the expression for $W_{1}$. We denote $\gamma
_{1}:=\overline{\gamma_{\overline{1}}}.$

\[l4\] If $(M,J,\theta)$ is an embeddable closed strictly pseudoconvex CR $3$-manifold with $c_{1}(T_{1,0}M)=0$, then there are $u\in C_{\mathbb{R}
}^{\infty}\left(  M\right)  $ and $\gamma=\gamma_{\overline{1}}\theta
^{\overline{1}}\in \Omega^{0,1}\left(  M\right)  $ with $$\gamma_{\overline{1},1}=\gamma_{1,\overline{1}}=0$$ such that $$W_{1}=2P_{1}u+i\left(  A_{11}\gamma_{\overline{1}}-\gamma_{1,0}\right)
.\label{0}$$

By choosing $\eta=\sigma_{\overline{1}}\theta^{\overline{1}}$ as in Lemma \[l3\], where $\sigma$ is chosen from Lemma \[l2\],** **there are$$\varphi=u+iv\in C_{\mathbb{C}
}^{\infty}\left(  M\right)$$ and$$\gamma=\gamma_{\overline{1}}\theta^{\overline{1}}\in \Omega^{0,1}\left(
M\right)  \cap \ker \left(  \square_{b}\right)$$ such that$$\sigma_{\overline{1}}=\varphi_{\overline{1}}+\gamma_{\overline{1}}\label{18}$$

Note that$$\square_{b}\gamma=0\Longrightarrow \overline{\partial}_{b}^{\ast}\gamma=0\Longrightarrow \gamma_{\overline{1},1}=0\label{2}$$ and$$\sigma_{1}=\left(  \overline{\varphi}\right)  _{1}+\gamma_{1}.\label{19}$$

Thus$$\begin{array}
[c]{ccl}\sigma_{1,\overline{1}1} & = & (\overline{\varphi})_{,1\overline{1}1}+\gamma_{1,\overline{1}1}\text{ \ }\mathrm{by}\text{ }\left(  \ref{19}\right)
\\
& = & (\overline{\varphi})_{,1\overline{1}1}\text{ \ }\mathrm{by}\text{
}\left(  \ref{2}\right) \\
& = & (\overline{\varphi})_{,\overline{1}11}+i(\overline{\varphi})_{,01}\text{
