Painleve Test and the Resolution of Singularities for Integrable Equations

We prove that under a very general setting, a system of ODE passes the Painleve test if and only if there is a good change of variable, such that the pole singularity solutions are converted to regular power series, while the converted ODE system is still kept regular. A consequence is that all principal balances of an ODE system converge. We also prove that the results are natural with respect to Hamiltonian systems.