Monomial Rota-Baxter operators of nonzero weight on $F[x,y]$ coming from averaging operators

The intensive study of Rota-Baxter operators on the polynomial algebra $F[x]$ has been started with the work of S.H. Zheng, L. Guo, and M. Rosenkranz (2015). We deal with the case of two variables and monomial Rota-Baxter operators of nonzero weight. The family of such operators arisen from homomorphic averaging operators on $F[x,y]$ is described.