Generic degeneracies in two-parameter families of vector fields on surfaces

Let a generic two-parameter family of vector fields be given on a surface. The degeneracies that can occur are described in terms of the degeneracies in Sotomayor's theorem and a new finite list.

Generic two-parameter degeneracy conjecture. In generic two-parameter families of vector fields on a surface, only the following degeneracies may occur: one or two simultaneous degeneracies from the list in Sotomayor's theorem, or exactly one degeneracy from a new finite list.

This is the proposed analogue, for two-parameter families, of Sotomayor's classification of degeneracies in generic one-parameter families. The statement remains a general goal of the authors; the preceding transversality heuristic is not yet a rigorous theorem in the relevant infinite-dimensional setting.