If (U, Ω) is a paired set, then for every element u in U , we denote by u e the element that belongs to the same pair as u and is distinct from u. Consider a pair of 2-matroids, Q 1 = (U, Ω, r 1 ) and Q 2 = (U, Ω, r 2 ), defined on the same partitioned set (U, Ω). Set R = r 1 + r 2 . Then P = (U, Ω, R) is a parity system, which we call the sum of Q 1 and Q 2 . Furthermore a solution to Problem 3.8 for P will give rise to a solution to Problem 2.7 for Q 1 and Q 2 . We shall be especially interested in the case when P is a sum of a 2-matroid Q = (U, Ω, r) and the converse 2-matroid Q e = (U, Ω, r), where e r e is defined by the relation r(A) = e r( A), for every subtransversal e A of Ω. In this case ν(P ) = | Ω | if and only if Q has two complementary bases. Thus Problem 3.8