Let $f=f(x,y) \in \mathbb{C}[x,y]$ and denote $F_{\lambda}:=fy-\lambda$.
Assume that for infinitely many $\mathbb{C} \ni \lambda$'s, $F_{\lambda}$ is irreducible in $\mathbb{C}[x,y]$, hence $(F_{\lambda})$ is a prime ideal, and then $\mathbb{C}[x,y]/(F_{\lambda})$ is an integral domain, so it has a field of fractions, which we will denote by $Q_{\lambda}:=\operatorname{Frac}(\mathbb{C}[x,y]/(F_{\lambda}))$.
Is it possible to characterize all $f \in \mathbb{C}[x,y]$ such that $Q_{\lambda}$ is isomorphic to the field $\mathbb{C}(t)$, $t$ is a variable, for infinitely many $\mathbb{C} \ni \lambda$'s?
(From those $\lambda$'s for which $F_{\lambda}$ is irreducible, since we which to guarantee that $\mathbb{C}[x,y]/(F_{\lambda})$ is an integral domain).
Remarks:
(1) The following is a quotation from the answer of user26857 to this question: Let $A$ be a UFD, and $f\in A[Y]$ irreducible with $\deg f\ge 1$. Then the field of fractions of $A[Y]/(f)$ is $K[Y]/(f)$, where $K$ is the field of fractions of $A$. Here, in my notations: $A=\mathbb{C}[x]$ $f=F_{\lambda}$, so $Q_{\lambda}=\mathbb{C}(x)[y]/(F_{\lambda})$.
(2) See also this question.
Thank you very much!