Let $C$ be a generic genus 1 curve embedded in $P^1\times P^1$ as a (2,2)-divisor.

Each projection defines $C$ as a double cover of $P^1$ and induces an involution $\tau_i:C\to C$.

Let $G\simeq \mathbb Z/2 * \mathbb Z/2$ be the subgroup of $Aut(C)$ generated by these. Note that $G$ is infinite.

My question is: are there points on $C$ with finite orbit under $G$?