For a given set $S\subseteq \mathbb {Z}_m$ and $\overline {n}\in \mathbb {Z}_m$ , let $R_S(\overline {n})$ denote the number of solutions of the equation $\overline {n}=\overline {s}+\overline {s'}$ with ordered pairs $(\overline {s},\overline {s'})\in S^2$ . We determine the structure of $A,B\subseteq \mathbb {Z}_m$ with $|(A\cup B)\setminus (A\cap B)|=m-2$ such that $R_{A}(\overline {n})=R_{B}(\overline {n})$ for all $\overline {n}\in \mathbb {Z}_m$ , where m is an even integer.

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对于给定的集合$S\subseteq \mathbb {Z}_m$和$\overline {n}\in \mathbb {Z}_m$， 让$R_S(\overline {n})$表示方程的解数$\overline {n}=\overline {s}+\overline {s'}$有序对$(\overline {s},\overline {s'})\in S^2$. 我们确定结构$A,B\subseteq \mathbb {Z}_m$和$|(A\cup B)\setminus (A\cap B)|=m-2$这样$R_{A}(\overline {n})=R_{B}(\overline {n})$对所有人$\overline {n}\in \mathbb {Z}_m$， 在哪里米是偶数。