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

Okamoto's space for the first Painlev\'e equation in Boutroux coordinates