We make a theoretical study of the $eta(1405) to pi^{0} f_0(980)$ and $eta(1405) to pi^{0} a_0(980)$ reactions with an aim to determine the isospin violation and the mixing of the $f_0(980)$ and $a_0(980)$ resonances. We make use of the chiral unitary approach where these two resonances appear as composite states of two mesons, dynamically generated by the meson-meson interaction provided by chiral Lagrangians. We obtain a very narrow shape for the $f_0(980)$ production in agreement with a BES experiment. As to the amount of isospin violation, or $f_0(980)$ and $a_0(980)$ mixing, assuming constant vertices for the primary $eta(1405)rightarrow pi^{0}Kbar{K}$ and $eta(1405)rightarrow pi^{0}pi^{0}eta$ production, we find results which are much smaller than found in the recent experimental BES paper, but consistent with results found in two other related BES experiments. We have tried to understand this anomaly by assuming an I=1 mixture in the $eta(1405)$ wave function, but this leads to a much bigger width of the $f_0(980)$ mass distribution than observed experimentally. The problem is solved by using the primary production driven by $eta to K^* bar K$ followed by $K^* to K pi$, which induces an extra singularity in the loop functions needed to produce the $f_0(980)$ and $a_0(980)$ resonances. Improving upon earlier work along the same lines, and using the chiral unitary approach, we can now predict absolute values for the ratio $Gamma(pi^0, pi^+ pi^-)/Gamma(pi^0, pi^0 eta)$ which are in fair agreement with experiment. We also show that the same results hold if we had the $eta(1475)$ resonance or a mixture of these two states, as seems to be the case in the BES experiment.
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