  By describing the $X(3872)$ using the extended Friedrichs scheme, in which
$D\bar D^*$ is the dominant component, we calculate the decay rates of the
$X(3872)$ to $\pi^0$ and a $P$-wave charmonium $\chi_{cJ}$ state with $J=0,1$,
or $2$, and its decays to $J/\psi\pi^+\pi^-$ where $\pi^+\pi^-$ are assumed to
be produced via an intermediate $\rho$ state. The decay widths of
$X(3872)\to\chi_{cJ}\pi^0$ for $J=0,1,2$ are of the same order. However, this
model calculation exhibits that the decay rate of $X(3872)$ to $\chi_{c1}\pi^0$
is one order of magnitude smaller than its decay rate to $J/\psi\pi^+\pi^-$.
