Filtered Lie conformal algebras whose associated graded algebras are isomorphic to that of general conformal algebra $g{c}_1$

Let $G$ be a filtered Lie conformal algebra whose associated graded conformal algebra is isomorphic to that of general conformal algebra $g{c}_1$. In this paper, we prove that $G\cong g{c}_1$ or $\mathrm{g}\mathrm{r}g{c}_1$ (the associated graded conformal algebra of $g{c}_1$), by making use of some results on the second cohomology groups of the conformal algebra $\text{fg}$ with coefficients in its module $M_{b,0}$ of rank 1, where $\text{fg}=\text{Vir}⋉M_{a,0}$ is the semi-direct sum of the Virasoro conformal algebra $\text{Vir}$ with its module $M_{a,0}$. Furthermore, we prove that $\mathrm{g}\mathrm{r}g{c}_1$ does not have a nontrivial representation on a finite $\text{C}[\mathrm{\partial }]$-module, this provides an example of a finitely freely generated simple Lie conformal algebra of linear growth that cannot be embedded into the general conformal algebra $g{c}_N$ for any $N$.