In this paper, we study the structure of simple canonical Nambu 3-Lie algebra $$A_{\partial }=\sum \nolimits _{m\in Z} F z\exp (mx) \oplus \sum \nolimits _{m\in Z}F y\exp (mx)$$ A ∂ = ∑ m ∈ Z F z exp ( m x ) ⊕ ∑ m ∈ Z F y exp ( m x ) . We pay close attention to a special class of Rota–Baxter operators, which are k-order homogeneous Rota–Baxter operators R of weight 1 and weight 0 satisfying $$R(L_m)=f(m+k)L_{m+k}$$ R ( L m ) = f ( m + k ) L m + k , $$R(M_m)=g(m+k)M_{m+k}$$ R ( M m ) = g ( m + k ) M m + k for all generators $$\{ L_m=z\exp (mx),$$ { L m = z exp ( m x ) , $$ M_m= y\exp (-mx)~~| ~~m\in Z\}$$ M m = y exp ( - m x ) | m ∈ Z } , where $$f, g : A_{\partial } \rightarrow F$$ f , g : A ∂ → F are functions and $$k\in Z$$ k ∈ Z . We obtain that R is a k-order homogeneous Rota–Baxter operator on $$A_{\partial }$$ A ∂ of weight 1 with $$k\ne 0$$ k ≠ 0 if and only if $$R=0$$ R = 0 , and R is a 0-order homogeneous Rota–Baxter operator on $$A_{\partial }$$ A ∂ of weight 1 if and only if R is one of the ten possibilities described in Theorems 2.4 and 2.8; R is a k-order homogeneous Rota–Baxter operator on $$A_{\partial }$$ A ∂ of weight 0 with $$k\ne 0$$ k ≠ 0 if and only if R satisfies Theorem 3.1; and R is a 0-order homogeneous Rota–Baxter operator on $$A_{\partial }$$ A ∂ of weight 0 if and only if R is one of the four possibilities described in Theorem 3.3