Classification of Homogeneous Odd Rota--Baxter Operators on a Modified Witt-Type Lie Superalgebra

We classify all homogeneous odd (i.e., parity-reversing) Rota--Baxter operators of weight zero on the modified Witt-type Lie superalgebra $W = \langle L_m, G_n \rangle_{m,n\in\Z}$. Our classification shows that nontrivial such operators are highly constrained: either $g \equiv 0$ and $f$ is arbitrary, or $g \not\equiv 0$ forces $f \equiv 0$, and $g$ must take one of several rigid forms dictated by the integer shift $k$ (necessarily odd when $g(0) \neq 0$). We prove that every Rota--Baxter operator on $W$ decomposes uniquely into even and odd homogeneous components; we restrict our attention to the odd case, which yields the full nontrivial structure. Furthermore, we show that all derivations of $W$ are inner, that no Rota--Baxter operator on $W$ is invertible, and we describe the induced super pre-Lie algebra structure together with its cohomological interpretation.