  The main purpose of this paper is to introduce a new class of Hamiltonian
scattering systems of the cone potential type that can be integrated via the
asymptotic velocity. For a large subclass, the asymptotic data of the
trajectories define a global canonical diffeomorphism $A$ that brings the
system into the normal form $\dot P=0$, $\dot Q=P$.
  The integrability theory applies for example to a system of $n$ particles on
the line interacting pairwise through rather general repulsive potentials. The
inverse $r$-power potential for arbitrary $r>0$ is included, the reduction to
normal form being carried out for the exponents $r>1$. In particular, the
Calogero system is obtained for $r=2$. The treatment covers also the
nonperiodic Toda lattice.
  The cone potentials that we allow can undergo small perturbations in any
arbitrary compact set without losing the integrability and the reduction to
normal form.
