  The heavy meson-antimeson pairs, where one is an excited $\bigl ({3 \over 2}
\bigr) ^+ $ meson and the other is a ground state $\bigl ({1 \over 2} \bigr)
^-$ meson, namely the pairs ($D_1(2420) \bar D$ + c.c.), ($D_1(2420) \bar D^*$
+ c.c.), ($D_2(2460) \bar D^*$ + c.c.) in the charm sector and ($B_1(5721) \bar
B$ + c.c.), ($B_1(5721) \bar B^*$ + c.c.), ($B_2(5747) \bar B^*$ + c.c.) in the
bottom sector, are allowed, by the quantum numbers, to be produced in the $S$
wave in $e^+e^-$ annihilation. We show, however, that such $S$-wave production
is forbidden by the heavy quark spin symmetry. Thus the yield of the considered
meson pairs in $e^+e^-$ annihilation should be significantly suppressed near
the respective thresholds. In our view, this substantially weakens the
arguments for considering the Y(4260) charmonium-like resonance as a $D_1 \bar
D$ molecular state.
