Partitions of Zm with identical representation functions

For any positive integer $m$, let $Z_m$ be the set of residue classes modulo $m$. For $A\subseteq Z_m$ and $\overline{n}\in Z_m$, let the representation function $R_A\left(\overline{n}\right)$ 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 $m=2^{\alpha }M$, where $\alpha$ is a nonnegative integer and $M$ is a positive odd integer. In this paper, we prove that if $M=1$ and $2\nmid \alpha$, then there exist two distinct sets $A,B\subseteq Z_m$ with $A\cup B=Z_m\setminus \left\{\overline{r_1}\right\},A\cap B=\left\{\overline{r_2}\right\}$ such that $R_A\left(\overline{n}\right)=R_B\left(\overline{n}\right)$ for all $\overline{n}\in Z_m$. We also prove that if $M\ge 3$ or $M=1$ and $2\mid \alpha$, then there do not exist two distinct sets $A,B\subseteq Z_m$ with $A\cup B=Z_m\setminus \left\{\overline{r_1}\right\}$ and $A\cap B=\left\{\overline{r_2}\right\}$ such that $R_A\left(\overline{n}\right)=R_B\left(\overline{n}\right)$ for all $\overline{n}\in Z_m$