  Propagation of particles with emission of arbitrary number of identical
bosons all being at rest is considered. It is shown that in certain models the
tree-level amplitudes for production of $n$ scalar bosons by two incoming
particles are all equal to zero at the threshold starting from some small
number $n$. In particular this nullification occurs for production of massive
scalars by two Goldstone bosons in the linear sigma model for $n > 1$ and also
for production of Higgs bosons in the Standard Model by gauge bosons and/or by
fermions, provided that the ratia of their masses to that of the Higgs boson
take special discrete values.
