  The model of the pure one-pion exchange mechanism, which gives a good
description of the GAMS results on the alteration of the $S$-wave $\pi^0\pi^0$
mass spectrum in the $f_0(980)$ region in the reaction $\pi^-p\to\pi^0\pi^0n$
with increasing $-t$, is compared with the recent detailed data on the $m$ and
$t$ distributions of the $\pi^-p\to\pi^0\pi^0n$ events obtained by the BNL-E852
Collaboration. It is shown that the predictions of this model are not confirmed
by the BNL data. Therefore the observed phenomenon should be explained by the
different exchange mechanism. It is most likely to be the $a_1$ exchange
mechanism.
