Properties of the Absolute That Affect Smoothness of Flows on Closed Surfaces

Let $M^2_g$ be a closed orientable surface of genus $g\ge2$, endowed with the structure of a Riemann manifold of constant negative curvature. For the universal covering $\Delta$, there is the notion of absolute, each of whose points determines an asymptotic direction of a bundle of parallel equidirected geodesics. In the paper it is proved that there is a set $U_g$ on the absolute having the cardinality of the continuum and such that if an arbitrary flow on $M^2_g$ has a semitrajectory whose covering has asymptotic direction defined by a point from $U_g$, then this flow is not analytical and has infinitely many stationary points.