Let $C$ be a (very) general genus 1 curve embedded in $\mathbb{CP}^1\times \mathbb{CP}^1$ as a (2,2)-divisor.

Each projection defines $C$ as a double cover of $\mathbb{CP}^1$ 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.

Are there points on $C$ with finite orbit under $G$?