  We study the completeness and connectedness of asymptotic behaviours of
solutions of the first Painlev\'e equation $\op{d}^2y/\op{d}x^2=6\, y^2+ x$, in
the limit $x\to\infty$, $x\in\C$. This problem arises in various physical
contexts including the critical behaviour near gradient catastrophe for the
focusing nonlinear Schr\"odinger equation. We prove that the complex limit set
of solutions is non-empty, compact and invariant under the flow of the limiting
autonomous Hamiltonian system, that the infinity set of the vector field is a
repellor for the dynamics and obtain new proofs for solutions near the
equilibrium points of the autonomous flow. The results rely on a realization of
Okamoto's space, i.e., the space of initial values compactified and regularized
by embedding in $\C\Proj 2$ through an explicit construction of nine blow-ups.
