Global parametrizations of the certain real algebraic surface

The certain real algebraic variety $\Omega$ in $\mathbb{R}^3$ is considered. This variety plays an important role in investigation of the normalized Ricci flow on generalized Wallach spaces. The method for constructing of the global parametric representation of this variety with the help of elimination theory and computer algebra methods is proposed. Parametrization of the discriminant set of a real cubic polynomial is used. Three different parametrizations of the variety $\Omega$ are obtained, each of which can be used out of some critical values of parameters.