For any positive integer m, let ℤm be the additive group of residue classes modulo m. For A ⊆ ℤm and \(\overline{n}\in\mathbb{Z}_{m}\) , let the representation function \(R_{A}(\overline{n})\) denote the number of solutions of the equation \(\overline{n}=\overline{a}+\overline{a^{\prime}}\) with unordered pairs \((\overline{a},\overline{a^{\prime}})\in A\times A\) . Let m = 2αM > 2, where α is a positive integer and M is a positive odd integer. In this paper, the author proves that if M ≥ 3, then there exist two distinct sets A, B ⊆ ℤm with ∣A ∪ B∣ = m − 2, A ∩ B = ∅ and \(B\ne{\overline{m}\over{2}}+A\) such that \(R_{A}(\overline{n})=R_{B}(\overline{n})\) for all \(\overline{n}\in\mathbb{Z}_{m}\) . The author also proves that if M = 1 and A, B ⊆ ℤm with ∣A ∪ B∣ = m − 2 and A ∩ B = ∅, then \(R_{A}(\overline{n})=R_{B}(\overline{n})\) for all \(\overline{n}\in\mathbb{Z}_{m}\) if and only if \(B={\overline{m}\over{2}}+A\) .
