Abstract We improve the Painlevé test in such a way that negative resonances can be treated. To this end we demand that the general solution of both the given nonlinear equation and its linearisation be single valued. This gives rise to compatibility conditions for every resonance. For equations with no principal branch, such as Chazy's equation, but with enough integer resonances, we (formally) build the general solution with an essential singularity, reducing to the Painlevé (finite pole) solutions for special choices of arbitrary constant. In the context of integrable hierarchies, our approach gives a clear relationship between negative resonances and lower order commuting flows.