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The isospin-breaking decay $X(3872)\to(D^*\bar D+ \bar D^*D) \to\pi^0D\barD\to\pi^0\pi^+\pi^-$ is discussed. In its amplitude there is a trianglelogarithmic singularity, due to which the dominant contribution to$BR(X(3872)\to\pi^0\pi^+\pi^-)$ comes from the production of the $\pi^+\pi^-$system in a narrow interval of the invariant mass $m_{\pi^+\pi^-}$ near thevalue of $2m_{D^0} \approx 3.73$ GeV. The analysis shows that$BR(X(3872)\to\pi^0 \pi^+\pi^-)$ can be expected at the level of$10^{-3}$--$10^{-4}$. This estimate includes, in particular, the assumptionthat the $S$-wave inelastic scattering length $|\alpha''_{D^0\bar D^0\to\pi^+\pi^-}|\approx1/(2m_{D^{*+}})\approx0.25\ \mbox{GeV}^{-1}$.

Decay $X(3872)\to\pi^0\pi^+\pi^-$ and $S$-wave $D^0\bar D^0 \to\pi^+\pi^-$ scattering length