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  \(U_3(F)\) $U_3(F)$ 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  \(U_3(F)\) $U_3(F)$ , and computation with the help of Singular of the Gröbner basis for the system of equations arisen on the coefficients of R.