Following a result of Bombieri, Masser, and Zannier on tori, the second author proved that the intersection of a transversal curve C in a power Eg of a C. M. elliptic curve with the union of all algebraic subgroups of Eg of codimension 2 is finite. Here transversal means that C is not contained in any translate of an algebraic subgroup of codimension 1. We merge this result with Faltings' theorem that C∩Γ is finite when Γ is a finite rank subgroup of Eg. We obtain the finiteness of the intersection of C with the union of all Γ+B for Ban abelian subvariety of codimension 2. As a corollary, we generalize the previous result to a curve C not contained in any proper algebraic subgroup, but possibly contained in a translate. We also have weaker analog results in the non C. M. case.