let $(x, t^{1, 0}x)$ be a compact cr manifold and $(l, h)$ be a hermitian cr line bundle over $x$. when $x$ is levi-flat and $l$ is positive, ohsawa and sibony constructed for every $\kappa\in\mathbb n$ a cr projective embedding of $c^\kappa$-smooth of the levi-flat cr manifold. adachi constructed a counterexample to show that the $c^k$-smooth can not be generalized to $c^\infty$-smooth. the difficulty comes from the fact that the kohn laplacian is not hypoelliptic on levi flat manifolds. in this talk, we will consider cr manifold $x$ with a transversal cr $g$-action where $g$ is a compact lie group and $g$ can be lifted to a cr line bundle $l$ over $x$. the talk will be divided into two parts. in the first part, we will talk about the morse inequalities for the fourier components of kohn-rossi cohomology on cr manifolds with transversal cr $s^1$-action. by studying the partial szeg\"o kernel on $(0