  We consider molecules made of two one-dimensional short-range-interacting
bosonic atoms. We show that in the process of scattering of these molecules off
a narrow barrier, odd incident waves produce \emph{no unbound atoms, even when
the incident energy exceeds the dissociation threshold}. This effect is a
consequence of a prohibition on chemical reactions acting in a generally
unphysical Bethe Ansatz integrable system of a $C_{2}$-type, with which our
system shares the spatially odd eigenstates. We suggest several experimental
implementations of the effect. We also propose to use the monomer production as
an alternative read-out channel in an atom interferometer: unlike in the
standard interferometric schemes, no spatial separation of the output channels
will be required.
