strictly pseudoconvex CR $3$-manifold $(M,J,\theta)$ with vanishing pseudohermitian torsion is embeddable. As a consequence, we are able to show that one of existence theorems for the pseudo-Einstein contact form in this paper.

\[c2\] Let $(M,J,\theta)$ be a closed strictly pseudoconvex CR $3$-manifold with $c_{1}(T_{1,0}M)=0$. Then $M$ admits a globally defined pseudo-Einstein contact form $e^{\frac{\left(  f+2u\right)  }{3}}\theta$ for any CR-pluriharmonic function $f$ if the pseudohermitian torsion is vanishing (i.e. Sasakian). More precisely, we have $$\gamma_{1,0}=0.$$

Note that we do not know whether it holds that $\gamma=0$ in the situation as in Corollary \[c2\]. However, by deriving the Bôchner-type estimate as in the Lemma \[l8\], we have$${\displaystyle \int \limits_{M}}
\left(  2R-Tor\right)  \left(  \gamma,\gamma \right)  d\mu+2{\displaystyle \int \limits_{M}}
\left \vert \gamma_{1,1}\right \vert ^{2}d\mu+\frac{1}{2}{\displaystyle \int \limits_{M}}
\left(  P_{0}f\right)  fd\mu=0$$ if $\widetilde{\theta}=e^{\frac{\left(  f+2u\right)  }{3}}\theta$ is a pseudo-Einstein contact form. It will conclude $$\gamma=0$$ under certain pseudohermitian geometric assumptions and obtain the solvability of the inhomogeneous tangential Cauchy-Riemann equation (\[2018A\]). Hereafter, we will use the result in [@ccy] implicitly, which says that if $(M,J,\theta)$ is a closed strictly pseudoconvex CR $3$-manifold with nonnegative CR Paneitz operator and positive Tanaka-Webster scalar curvature, then $(M,J,\theta)$ is embeddable. This ensures the embeddability in the hypothesis holds for closed strictly 