Nonlocality and scattering transform for for multidimensional dispersionless systems. Some connections with integral geometry.

At the present time the class of dispersionless integrable systems attracts serious attention. Manakov and Santini suggested a modification of the scattering transform for vector fields, which is appropriate for integration of multidimensional dispersionless systems, but it requires seriuos additional work to make it sufficiently strict. We provide first strict results in this area.