Partitions of nonnegative integers with identical representation functions

Let $\mathbb{N}$ be the set of all nonnegative integers. For any integer $r$ and $m$, let $r+m\mathbb{N}=\{r+mk:k\in \mathbb{N}\}$. For $S\subseteq \mathbb{N}$ and $n\in \mathbb{N}$, let $R_S(n)$ denote the number of solutions of the equation $n=s+s^{\mathrm{\prime }}$ with $s,s^{\mathrm{\prime }}\in S$ and $s<s^{\mathrm{\prime }}$. Let $r_1,r_2,m$ be integers with $0<r_1<r_2<m$ and $2\mid r_1$. In this paper, we prove that there exist two sets $C$ and $D$ with $C\cup D=\mathbb{N}$ and $C\cap D=(r_1+m\mathbb{N})\cup (r_2+m\mathbb{N})$ such that $R_C(n)=R_D(n)$ for all $n\in \mathbb{N}$ if and only if there exists a positive integer $l$ such that $r_1=2^{2l+1}-2,r_2=2^{2l+1}-1,m=2^{2l+2}-2$.