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In the paper we study homogeneous Rota-Baxter operators with weight zero onthe infinite dimensional simple $3$-Lie algebra $A_{\omega}$ over a field $F$ ($ch F=0$ ) which is realized by an associative commutative algebra $A$ and aderivation $\Delta$ and an involution $\omega$ ( Lemma \mref{lem:rbd3} ). Ahomogeneous Rota-Baxter operator on $A_{\omega}$ is a linear map $R$ of$A_{\omega}$ satisfying $R(L_m)=f(m)L_m$ for all generators of $A_{\omega}$,where $f : A_{\omega} \rightarrow F$. We proved that $R$ is a homogeneousRota-Baxter operator on $A_{\omega}$ if and only if $R$ is the one of the fivepossibilities $R_{0_1}$, $R_{0_2}$,$R_{0_3}$,$R_{0_4}$ and $R_{0_5}$, which aredescribed in Theorem \mref{thm:thm1}, \mref{thm:thm4}, \mref{thm:thm01},\mref{thm:thm03} and \mref{thm:thm04}. By the five homogeneous Rota-Baxteroperators $R_{0_i}$, we construct new $3$-Lie algebras $(A, [ , , ]_i)$ for$1\leq i\leq 5$, such that $R_{0_i}$ is the homogeneous Rota-Baxter operator on$3$-Lie algebra $(A, [ , , ]_i)$, respectively.

Homogeneous Rota-Baxter operators on $3$-Lie algebra $A_{\omega}$