  We introduce a new rigorous method, based on Borel summability and asymptotic
constants of motion generalizing \cite{invent} and \cite{ode1}, to analyze
singular behavior of nonlinear ODEs in a neighborhood of infinity and provide
global information about their solutions. In equations with the
Painlev\'e-Kowalevski (P-K) property (stating that movable singularities are
not branched) it allows for solving connection problems. The analysis in
carried in detail for P$_1$, $y"=6y^2+z$, for which we find the Stokes
multipliers in closed form and global asymptotics for solutions having
power-like behavior in some direction in $\CC$, in particular for the
tritronqu\'ees. Calculating the Stokes multipliers solely relies on the P-K
property and does not use linearization techniques such as Riemann-Hilbert or
isomonodromic reformulations. We discuss how the approach would work to
calculate connection constants for a larger class of P-K integrable equations.
We develop methods for finding asymptotic expansions in sectors where solutions
have infinitely many singularities. These techniques do not rely on
integrability and apply to more general second order ODEs which, after
normalization, are asymptotically close to autonomous Hamiltonian systems.
