Collision of Orbits on an Elliptic Surface

Let $C$ be a smooth projective curve defined over $\Qbar$, let $\pi:\mathcal{E}\lra C$ 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 $\l\in C(\Qbar)$ with the property that for some nonzero integers $m_{1,\l},m_{2,\l}$, we have that $[m_{i,\l}](\sigma_{{P_{i}}}(\l))=\sigma_{Q}(\l)$ (for $i=1,2$) on the smooth fiber $E_\l$ of $\mathcal{E}$.