  The charmonium transitions from the $\chi_{cJ}$ states to $\eta_c$ with
emission of one or two pions are considered. It is shown that the only
processes of such type arising in the leading order in the QCD multipole
expansion are the decays $\chi_{c1} \to \eta_c \pi^+ \pi^-$ and $\chi_{c0} \to
\eta_c \pi^0$. The absolute rate of the latter decay, in spite of being
isotopically suppressed, is by about an order of magnitude larger than that of
the former, so that the branching fractions for the two processes are
approximately equal, and are expected to be significantly below the recent
upper bound reported by BESIII. The rates of both decays are also related to
that of another similar transition $h_c \to J/\psi \pi^0$.
