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