  The problem of scattering of particles on the line with repulsive
interactions, gives rise to some well-known integrable Hamiltonian systems, for
example, the nonperiodic Toda lattice or Calogero's system. The aim of this
note is to outline our researches which proved the integrability of a much
larger class of systems, including some that had never been considered, such as
the scattering with very-long-range interaction potential. The integrability of
all these systems survives any small enough perturbation of the potential in an
arbitrary compact set. Our framework is based on the concept of cone
potentials, as defined below, which include the scattering on the line as a
particular case.
