Let $Γ$ be an Abelian group. In this paper I characterize the $A$-paths of weight $0\inΓ$ that have the Erdős-Pósa property. Using this in an auxiliary graph, one can also easily characterize the $A$-paths of weight $γ\inΓ$ that have the Erdős-Pósa property. These results also extend to long paths, that is paths of some minimum length. A structural result on zero walls with non-zero linkages will be proven as a means to prove the main result of this paper. This immediately implies that zero cycles with respect to an Abelian group $Γ$ have the Erdős-Pósa property.