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Instablilty of pole singularities for the Chazy equation
We prove that the negative resonances of the Chazy equation (in the sense of Painleve analysis) can be related directly to its group-invariance properties. These resonances indicate in this case the instability of pole singularities. Depending on the value of a parameter in the equation, an unstable isolated pole may turn into the familiar natural boundary, or split into several isolated singularities. In the first case, a convergent series representation involving exponentially small corrections can be given. This reconciles several earlier approaches to the interpretation of negative resonances. On the other hand, we also prove that pole singularities with the maximum number of positive resonances are stable. The proofs rely on general properties of nonlinear Fuchsian equations.