  The well-known model of the triangle diagrams with $D^*\bar DD^*$ and $\bar
D^*D\bar D^*$ mesons in the loops is compared with the modern data on the
amplitude 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 a
parameter $\xi$ characterizing the scale of the isotopic symmetry violation in
this decay and find a lower limit of $\xi\simeq0.0916$. The model incorporates
the only fitted parameter associated with the form factor. We analyze in detail
the 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 amplitude
by the form factor increases, $\xi$ increases. Due to the fact that the
$X(3872)$ resonance is located practically at the threshold of the $D^0\bar
D^{*0}$ channel, the amplitude of $X(3872) \to\pi^0\chi_{c1}(1P)$ turns out to
be proportional to $\sqrt{m_d -m_u}$. Using the estimating values for the
coupling 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 diagrams
is in reasonable agreement with the available data. Apart from the difference
in the masses of neutral and charged charmed mesons, any additional exotic
sources of isospin violation in $X(3872)\to\pi^0\chi_{c1}(1P)$ (such as a
significant difference between the coupling constants $g_{XD^0\bar D^{*0}}$ and
$g_{XD^+ D^{*-}}$) are not required to interpret the data. This indirectly
confirms the isotopic neutrality of the $X(3872)$, which is naturally realized
for the $c\bar c$ state $\chi_{c1}(2P)$.
