  Let $S$ be a $3$-dimensional quantum polynomial algebra, and $f \in S_2$ a
central regular element. The quotient algebra $A = S/(f)$ 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 $C(A)$.
