  We study the behavior of a three-dimensional dynamical system with respect to
some set $S$ given in 3-dimensional euclidian space. Geometrically such a
system arises from the normalized Ricci flow on some class of generalized
Wallach spaces that can be described by a real parameter $a\in (0,1/2)$, as
for~$S$ it represents the set of invariant Riemannian metrics of positive
sectional curvature on the Wallach spaces. Establishing that $S$ is bounded by
three conic surfaces and regarding the normalized Ricci flow as an abstract
dynamical system we find out the character of interrelations between that
system and~$S$ for all $a\in (0,1/2)$. These results can cover some well-known
results, in particular, they can imply that the normalized Ricci flow evolves
all generic invariant Riemannian metrics with positive sectional curvature into
metrics with mixed sectional curvature on the Wallach spaces corresponding to
the cases $a\in \{1/9, 1/8, 1/6\}$ of generalized Wallach spaces.
