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

A class of integrable Hamiltonian systems including scattering of particles on the line with repulsive interactions