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We study the isospin breaking in the reactions $\chi_{c1} \to \pi^0 \pi^+\pi^-$ and $\chi_{c1} \to \pi^0 \pi^0 \eta$ and its relation to the $a_0(980) -f_0(980)$ mixing, which was measured by the BESIII Collaboration. We show thatthe same theoretical model previously developed to study the $\chi_{c1} \to\eta \pi^+ \pi^-$ reaction (also measured by BESIII), and further explored inthe predictions to the $\eta_{c} \to \eta \pi^+ \pi^-$, can be successfullyemployed in the present study. We assume that the $\chi_{c1}$ behaves as an$SU(3)$ singlet to find the weight in which trios of pseudoscalars are created,followed by the final state interaction of pairs of mesons to describe how the$a_0(980)$ and $f_0(980)$ are dynamically generated, using the chiral unitaryapproach in coupled channels. The isospin violation is introduced through theuse of different masses for the charged and neutral kaons, either in thepropagators of the pairs of mesons created in the $\chi_{c1}$ decay, or in thepropagators inside the $T$ matrix, constructed through the unitarization of thescattering and transition amplitudes of pairs of pseudoscalar mesons. We findthat violating isospin inside the $T$ matrix makes the $\pi^0\eta \to\pi^+\pi^-$ amplitude nonzero, which gives an important contribution and alsoenhances the effect of the $K\bar{K}$ term. Also, we find that in the totalamplitude the most important effect is the isospin breaking inside the $T$matrix, due to the constructive sum of $\pi^0\eta \to \pi^+\pi^-$ and $K\bar{K}\to \pi^+ \pi^-$, which is essential to get a good agreement with theexperimental measurement of the mixing.

$a_0(980) - f_0(980)$ mixing in $\chi_{c1} \to \pi^0 f_0(980) \to \pi^0 \pi^+ \pi^-$ and $\chi_{c1} \to \pi^0 a_0(980) \to \pi^0 \pi^0 \eta$