  The isospin violating $J/\Psi\to \phi \pi^0 \eta$ decay is thought to be
dominated by the mixing of the $f_0(980)$ and $a_0(980)$ scalar resonances. We
make a theoretical evaluation of the $J/\Psi\to \phi \pi^0 \eta$ decay
extending our own previous model for other $J/\Psi$ decays into one vector
meson and two pseudoscalars using the techniques of the chiral unitary
approach. The scalar resonances are dynamically generated through the final
state interaction of the pseudoscalar mesons implementing unitarity from the
lowest order ChPT amplitudes. Besides the direct $J/\Psi\phi PP$ vertex, other
mechanisms like the sequential exchange of vector and axial-vector mesons are
shown to be important in order to obtain the actual strength of the mixing. We
get a very good agreement with the $\pi^0 \eta$ invariant mass distribution and
branching ratio with recent BESIII data. Quantification of the
$f_0(980)-a_0(980)$ mixing is done and compared with results from several
experiments.
