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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.
Vsevolod Gubarev (Sat,) studied this question.