  The Poincar\'{e}-Bendixson theorem is one of the most fundamental tools to
capture the limit behaviors of orbits of flows. It was generalized and applied
to various phenomena in dynamical systems, differential equations, foliations,
group actions, translation lines, and semi-dynamical systems. On the other
hand, though the no-slip boundary condition is a fundamental condition in
differential equations and appears in various fluid phenomena, and Lakes of
Wada attractors naturally occur in discrete and continuous real dynamical
systems and complex dynamics, no generalizations of the Poincar\'{e}-Bendixson
theorem can be applied to any differential equations with no-slip boundary
condition on surfaces with boundary and flows with Lakes of Wada attractors. To
analyze them, we generalize the Poincar\'{e}-Bendixson theorem into one for
flows with arbitrarily many singular points on possibly non-compact surfaces by
introducing some concepts to describe limit behaviors and using methods of
foliation theory and general topology.
