On the behavior in the Euclidean or Lobachevsky plane of trajectories that cover trajectories of flows on closed surfaces. I

Consider a flow on a surface $M$ of nonpositive Euler characteristic whose set of equilibrium points can be deformed, in $M$, to a point (this, for example, is the case if there are only finitely many equilibrium points). For such a flow, it is proved that the semitrajectory of the covering flow on the universal cover (the Euclidean or Lobachevsky plane) of $M$ is either bounded or tends to infinity in a definite direction. For analytic flows (but not for $C^\infty$-flows), this conclusion holds without any conditions on the equilibrium points.
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