Rota-Baxter operators of weight zero on upper-triangular matrices of order three

We describe all Rota-Baxter operators $R$ of weight zero on the algebra $U_3(F)$ of upper-triangular matrices of order three over a field of characteristic 0. For this, we apply the following three ingredients: properties of $R(1)$, conjugation with suitable (anti)automorphisms of $U_3(F)$, and computation with the help of $\mathtt{\text{Singular}}$ of the Gröbner basis for the system of the equations arisen on the coefficients of $R$.