The E×CM component of Zilber–Pink for curves in A2

We prove the $E\times \mathrm{CM}$ component of Zilber–Pink for Hodge-generic algebraic curves in ${\mathcal{A}}_2$ over $\overline{\mathbb{Q}}$. Each such curve contains only finitely many points whose abelian surface is isogenous to a product of elliptic curves with at least one factor having complex multiplication.