  We prove that the negative resonances of the Chazy equation (in thesense of
Painlev\'e analysis) can be related directly to it sgroup-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.
