Algebraic functions with infinitely many values in a number field

We describe algebraic curves $X : F(x, y) = 0$ defined over $\overline{\mathbb{Q}}$ that satisfy the following property: there exist a number field $k$ and an infinite set $S \subset k$ such that, for every $y \in S$, the roots of the polynomial $F(x, y)$ belong to $k$.