Collision of Orbits on an Elliptic Surface

Let $C$ be a smooth projective curve defined over $\text{Qbar}$, let $\pi :\mathcal{E}\text{lra}C$ be an elliptic surface and let ${\sigma }_{P_1},{\sigma }_{P_2},{\sigma }_Q$ be sections of $\pi$ (corresponding to points $P_1,P_2,Q$ of the generic fiber $E$ of $\mathcal{E}$). We obtain a precise characterization, expressed solely in terms of the dynamical relations between the points $P_1,P_2,Q$ with respect to the endomorphism ring of $E$, so that there exist infinitely many $\text{ł}\in C(\text{Qbar})$ with the property that for some nonzero integers $m_{1,\text{ł}},m_{2,\text{ł}}$, we have that $[m_{i,\text{ł}}]({\sigma }_{P_i}(\text{ł}))={\sigma }_Q(\text{ł})$ (for $i=1,2$) on the smooth fiber $E_{\text{ł}}$ of $\mathcal{E}$.