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 U₃ (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 U₃ (F), and computation with the help of Singular of the Gr\"obner basis for the system of the equations arisen on the coefficients of R.