  Possible manifestations of the isospin-breaking mixing of the $a^0_0(980)$
and $f_0(980)$ resonances are analyzed in the $D^0\to K^0_S \pi^+\pi^-$ and
$D^0\to K^0_S\eta\pi^0$ decays. It is shown that the $a^0_0(980)-f_0(980)$
mixing has the most influence on the $\pi^+\pi^-$ mass spectrum in the decay
$D^0\to K^0_S\pi^+\pi^-$. Owing to the $a^0_0(980)-f_0(980)$ mixing, the
$f_0(980)$ peak in $D^0 \to K^0_Sf_0(980)\to K^0_S \pi^+\pi^-$ can experience
distortions comparable with its size. The effect essentially depends on the
relative phase between the amplitudes of the $D^0\to K^0_Sf_0(980)$ and $D^0\to
K^0_Sa^0_0(980)$ transitions.
