Linear-in-degree monomial Rota-Baxter of weight zero and averaging operators on $F[x,y]$ and $F_0[x,y]$

Rota-Baxter operators on the polynomial algebra have been actively studied since the work of S.H. Zheng, L. Guo, and M. Rosenkranz (2015). Monomial operators of an arbitrary weight (2016), as well as injective operators of weight zero on $F[x]$ (2021), have been described. The author described monomial Rota-Baxter operators of nonzero weight on $F[x,y]$ coming from averaging operators (2023) and studied the connection between monomial Rota-Baxter operators and averaging operators (2024). The main result of the current work is the classification of monomial Rota-Baxter operators of weight zero on $F[x,y]$ coming from monomial linear-in-degree averaging operators.