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