  We study the solutions of the second Painlev\'e equation in the space of
initial conditions first constructed by Okamoto, in the limit as the
independent variable, x, goes to infinity. Simultaneously, we study solutions
of the related equation known as the thirty-fourth Painlev\'e equation. By
considering degenerate cases of the autonomous flow, we recover the known
special solutions, which are either rational functions or expressible in terms
of Airy functions. We show that the solutions that do not vanish at infinity
possess an infinite number of poles. An essential element of our construction
is the proof that the union of exceptional lines is a repellor for the dynamics
in Okamoto's space. Moreover, we show that the limit set of the solutions
exists and is compact and connected.
