  We study the role of a possible nonet of light scalar mesons in the still
interesting $\eta \to 3\pi$ decay process, with the primary motivation of
learning more about the scalars themselves. The framework is a conventional
non-linear chiral Lagrangian of pseudoscalars and vectors, extended to include
the scalars. The parameters involving the scalars were previously obtained to
fit the s-wave $\pi\pi$ and $\pi$K scatterings in the region up to about 1 GeV
as well as the strong decay $\eta' \to \eta \pi\pi$. At first, one might expect
a large enhancement from diagrams including a light $\sigma(560)$. However
there is an amusing cancellation mechanism which prevents this from occurring.
In the simplest model there is an enhancement of about 13 per cent in the \e3p
decay rate due to the scalars. In a more complicated model which includes
derivative type symmetry breakers, the cancellation is modified and the scalars
contribute about 30 percent of the total decay rate (although the total is not
significantly changed). The vectors do not contribute much. Our model produces
a reasonable estimate for the related $a_0(980)-f_0(980)$ mixing strength,
which has been a topic of current debate. Promising directions for future work
along the present line are suggested.
