On investigation of the certain real algebraic surface

We provide the description of the certain real algebraic variety in $\mathbb{R}^3$. This variety plays an important role in investigation of Einstein's metrics, which evolution is studied by the normalized Ricci flow. We provide the description of the all singular points of the variety. All computations in the preprint were done with the help of computer algebra algorithms, in particular with the help of Gröbner basis and algorithms of polynomial ideals. We stated and proved an auxiliary result on the structure of the discriminant surface of the cubic polynomial.