  Recent experiments on $\eta\to 3\pi$ decays have provided an extremely
precise knowledge of the amplitudes across the Dalitz region which represent
stringent constraints on theoretical descriptions. We reconsider an approach in
which the low-energy chiral expansion is assumed to be optimally convergent in
an unphysical region surrounding the Adler zero, and the amplitude in the
physical region is uniquely deduced by an analyticity-based extrapolation using
the Khuri-Treiman dispersive formalism. We present an extension of the usual
formalism which implements the leading inelastic effects from the $K\bar{K}$
channel in the final-state $\pi\pi$ interaction as well as in the initial-state
$\eta\pi$ interaction. The constructed amplitude has an enlarged region of
validity and accounts in a realistic way for the influence of the two light
scalar resonances $f_0(980)$ and $a_0(980)$ in the dispersive integrals. It is
shown that the effect of these resonances in the low energy region of the $\eta
\to 3\pi$ decay is not negligible, in particular for the $3\pi^0$ mode, and
improves the description of the energy variation across the Dalitz plot. Some
remarks are made on the scale dependence and the value of the double quark mass
ratio $Q$.
