  We consider the tree amplitudes of production of $n_2$ scalar particles by
$n_1$ particles of another kind, where both initial and final particles are at
rest and on mass shell, in a model of two scalar fields with $O(2)$ symmetric
interaction and unequal masses. We find that these amplitudes are zero except
for the lowest possible $n_1$ and $n_2$, and that the cancellation of the
corresponding Feynman graphs occurs due to a special symmetry of the classical
mechanical counterpart of this theory. This feature is rather general and is
inherent in various other scalar field theories.
