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We consider the evaluation of the $\eta\pi$ isospin-violating vector andscalar form factors relying on a systematic application of analyticity andunitarity, combined with chiral expansion results. It is argued that the usualanalyticity properties do hold (i.e. no anomalous thresholds are present) inspite of the instability of the $\eta$ meson in QCD. Unitarity relates thevector form factor to the $\eta\pi \to \pi\pi$ amplitude: we exploit progressin formulating and solving the Khuri-Treiman equations for $\eta\to 3\pi$ andin experimental measurements of the Dalitz plot parameters to evaluate theshape of the $\rho$-meson peak. Observing this peak in the energy distributionof the $\tau\to \eta \pi \nu$ decay would be a background-free signature of asecond-class amplitude. The scalar form factor is also estimated from a phasedispersive representation using a plausible model for the $\eta\pi$ elasticscattering $S$-wave phase shift and a sum rule constraint in the inelasticregion. We indicate how a possibly exotic nature of the $a_0(980)$ scalar mesonmanifests itself in a dispersive approach. A remark is finally made on asecond-class amplitude in the $\tau\to\pi\pi\nu$ decay.

Analyticity of $\eta\pi$ isospin-violating form factors and the $\tau\to\eta\pi\nu$ second-class decay