Classification of noncommutative conics associated to symmetric regular superpotentials

Let $S$ be a $3$-dimensional quantum polynomial algebra, and $f\in S_2$ a central regular element. The quotient algebra $A=S\mathrm{/}(f)$ is called a noncommutative conic. For a noncommutative conic $A$, there is a finite dimensional algebra $C(A)$ which determines the singularity of $A$. In this paper, we mainly focus on a noncommutative conic such that its quadratic dual is commutative, which is equivalent to say, $S$ is determined by a symmetric regular superpotential. We classify these noncommutative conics up to isomorphism of the pairs $(S,f)$, and calculate the algebras $C(A)$.