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The well-known model of the triangle diagrams with $D^*\bar DD^*$ and $\barD^*D\bar D^*$ mesons in the loops is compared with the modern data on theamplitude of the $X(3872)\to\pi^0\chi_{c1}(1P)$ decay. Considering the$X(3872)$ object as a $\chi_{c1}(2P)$ charmonium state, we introduce aparameter $\xi$ characterizing the scale of the isotopic symmetry violation inthis decay and find a lower limit of $\xi\simeq0.0916$. The model incorporatesthe only fitted parameter associated with the form factor. We analyze in detailthe influence of the form factor on the amplitude $X(3872)\to\pi^0\chi_{c1}(1P)$ and on the parameter $\xi$. As the suppression of the amplitudeby the form factor increases, $\xi$ increases. Due to the fact that the$X(3872)$ resonance is located practically at the threshold of the $D^0\barD^{*0}$ channel, the amplitude of $X(3872) \to\pi^0\chi_{c1}(1P)$ turns out tobe proportional to $\sqrt{m_d -m_u}$. Using the estimating values for thecoupling constants $g_{XD\bar D^*}$, $g_{\chi_{ c1}D\bar D^*}$, and$g_{D^{*0}D^{*0}\pi^0}$, we show that the model of the triangle loop diagramsis in reasonable agreement with the available data. Apart from the differencein the masses of neutral and charged charmed mesons, any additional exoticsources of isospin violation in $X(3872)\to\pi^0\chi_{c1}(1P)$ (such as asignificant difference between the coupling constants $g_{XD^0\bar D^{*0}}$ and$g_{XD^+ D^{*-}}$) are not required to interpret the data. This indirectlyconfirms the isotopic neutrality of the $X(3872)$, which is naturally realizedfor the $c\bar c$ state $\chi_{c1}(2P)$.

Towards an estimate of the amplitude $X(3872)\to\pi^0\chi_{c1}(1P)$