  We study topological structures of the sets $(0,1/2)^3 \cap \Omega$ and
$(0,1/2)^3 \setminus \Omega$, where~$\Omega$ is one special algebraic surface
defined by a symmetric polynomial in variables $a_1,a_2,a_3$ of degree~$12$.
These problems arise in studying of general properties of degenerate singular
points of dynamical systems obtained from the normalized Ricci flow on
generalized Wallach spaces. Our main goal is to prove the connectedness of
$(0,1/2)^3 \cap \Omega$ and to determine the number of connected components of
$(0,1/2)^3 \setminus \Omega$.
  Key words and phrases: Riemannian metric, generalized Wallach space,
normalized Ricci flow, dynamical system, degenerate singular point of dynamical
system, real algebraic surface, singular point of real algebraic surface.
