Billiards on almost integrable polyhedral surfaces

The phase space of the geodesic flow on an almost integrable polyhedral surface is foliated into a one-parameter family of invariant surfaces. The flow on a typical invariant surface is minimal. We associate with an almost integrable polyhedral surface its holonomy group which is a subgroup of the group of motions of the Euclidean plane. We show that if the holonomy group is discrete then the flow on an invariant surface is ergodic if and only if it is minimal.