  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 \texttt{Singular} of the Gr\"{o}bner
basis for the system of the equations arisen on the coefficients of $R$.
