Near-integrability is usually associated with smooth small perturbations of smooth integrable systems. We show that studying integrable mechanical Hamiltonian flows with impacts that respect the symmetries of the integrable structure provide an additional rich class of non-smooth systems that can be studied by perturbation methods. Moreover, the analysis can be extended to systems with soft steep potentials that limit to the impact systems. For example, for some of these systems, we show that KAM theory may be applied, proving that for a large portion of phase space the perturbed motion is conjugate to rotations on a torus [1]. On the other hand, other simple impact systems have inherently non-rotational motion – we show cases in which the motion is conjugate to geodesic flow on a flat torus with several handles [2]. [1] M. Pnueli & V. Rom-Kedar “On near integrability of some impact systems”, SIAM-DS, 2018 to appear. [2] L. Becker, S. Elliott, B. Firester, S. Gonen Cohen, M. Pnueli & V. Rom-Kedar, in preparation.