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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 ofsolutions of the equation $n=s+s'$ with $s, s'\in S$ and $s<s'$. 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 apositive integer $l$ such that $r_{1}=2^{2l+1}-2, r_{2}=2^{2l+1}-1,m=2^{2l+2}-2$.

Partitions of nonnegative integers with identical representation functions