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

A direct method to find Stokes multipliers in closed form for P1 and more general integrable systems