Abstract
 I will report on results obtained with Duistermaat, Howes and Radnovic for the
completeness and connectedness of asymptotic behaviours of solutions of the
first, second and fourth Painlev\'e equations in the limit $x\to\infty$,
$x\in\mathbb C$. We prove that the complex limit set of solutions is nonempty,
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 $\mathbb
P^2$ through an explicit construction of nine blowups.
