\ }\mathrm{by}\text{ }\left(  \ref{6}\right) \\
& = & (\overline{\varphi})_{,\overline{1}11}+i\left[  (\overline{\varphi
})_{,10}+A_{11}(\overline{\varphi})_{,\overline{1}}\right]  \text{
\ }\mathrm{by}\text{ }\left(  \ref{6}\right)
\end{array}$$ and$$\sigma_{\overline{1},11}=\varphi_{,\overline{1}11}\text{ \ }\mathrm{from}\text{ }\left(  \ref{18}\right)  \text{ }\mathrm{and}\text{ }\left(
\ref{2}\right)$$ imply$$\begin{array}
[c]{ccl}W_{1} & = & R_{,1}-iA_{11,\overline{1}}\\
& = & \sigma_{\overline{1},11}+\sigma_{1,\overline{1}1}-i\sigma_{1,0}+iA_{11}\sigma_{\overline{1}}\text{ \ }\mathrm{by}\text{ }\left(
\ref{1}\right) \\
& = & \varphi_{,\overline{1}11}+(\overline{\varphi})_{,\overline{1}11}+iA_{11}(\overline{\varphi})_{,\overline{1}}-i\gamma_{1,0}+iA_{11}\left(
\varphi_{\overline{1}}+\gamma_{\overline{1}}\right) \\
& = & 2\left(  u_{\overline{,1}11}+iA_{11}u_{\overline{1}}\right)  +i\left(
A_{11}\gamma_{\overline{1}}-\gamma_{1,0}\right) \\
& = & 2P_{1}u+i\left(  A_{11}\gamma_{\overline{1}}-\gamma_{1,0}\right)
\end{array}
.$$

This completes the proof.

Now we are ready to give the proof of Theorem \[t1\]:

(Proof of Theorem \[t1\]**)** Set$$\widetilde{\theta}=e^{2\lambda}\theta \text{.}$$ By the transformation law (refer to Lemma 5.4 in [@h] or Lemma 3.1 in [@cw]), we know$$\widetilde{W_{1}}=e^{-3\lambda}\left(  W_{1}-6P_{1}\lambda \right)  ,\label{5}$$ where the notation with “tilde” means such quantity corresponds to the new contact form $\widetilde{\theta}$. With the help of $\left(  \ref{5}\right)  $ and Lemma \[l4\], we have$$\widetilde{W_{1}}=0$$ if and only if$$W_{1}=6P_{1}\lambda$$ if and only if $$6P_{1}\lambda=2P_{1}u+i\left(  A_{11}\gamma_{\overline{1}}-\gamma
_{1,0}\right)  .$$ That is to say $$P_{1}f=i\left(  A_{11}\gamma_{\overline{1}}-\gamma_{1,0}\right)$$ for $$f=\left(  6\lambda-2u\right)  .$$

From $\left(  \ref{7}\right)  $ and $$\gamma_{1,0\overline{1}}=\gamma_{1,\overline{1}0}+A_{\overline{1}\overline{1}}\gamma_{1,1}+A_{\overline{1}\overline{1},1}\gamma_{1}=A_{\overline
{1}\overline{1}}\gamma_{1,1}+A_{\overline{1}\overline{1},1}\gamma_{1},$$ we could deduce $f$ satisfies the fourth-order partial differential equation $$P_{0}f=2i\left[  \left(  A_{11}\gamma_{\overline{1}}\right)  _{,\overline{1}}-\left(  A_{\overline{1}\overline{1}}\gamma_{1}\right)  _{,1}\right]
\label{24}$$ where $P_{0}$ is the CR Paneitz  operator (see section$\ 2$). This suggests us there is an obstruction to the existence of pseudo-Einstein contact form 