  In this paper we study $k$-order homogeneous Rota-Baxter operators with
weight $1$ on the simple $3$-Lie algebra $A_{\omega}$ (over a field of
characteristic zero), which is realized by an associative commutative algebra
$A$ and a derivation $\Delta$ and an involution $\omega$ (Lemma
\mref{lem:rbd3}). A $k$-order homogeneous Rota-Baxter operator on $A_{\omega}$
is a linear map $R$ satisfying $R(L_m)=f(m+k)L_{m+k}$ for all generators $\{
L_m~ |~ m\in \mathbb Z \}$ of $A_{\omega}$ and a map $f : \mathbb Z
\rightarrow\mathbb F$, where $k\in \mathbb Z$. We prove that $R$ is a $k$-order
homogeneous Rota-Baxter operator on $A_{\omega}$ of weight $1$ with $k\neq 0$
if and only if $R=0$ (see Theorems 3.2, and $R$ is a $0$-order homogeneous
Rota-Baxter operator on $A_{\omega}$ of weight $1$ if and only if $R$ is one of
the forty possibilities which are described in Theorems3.5, 3.7, 3.9, 3.10,
3.18, 3.21 and 3.22.
