Peixotoʼs Theorem for vector fields on S2S2 with impasse points

The structural stability of vector fields with impasse regular curves on S2S2 is studied and a version of Peixotoʼs Theorem is established. Moreover a global analysis of normal forms of the constrained systemsA(x).x˙=F(x),x∈R3,A∈M(3),F:R3→R3 in the Poincaré ball (i.e. in the compactification of R3R3 with the sphere S2S2 of the infinity) is made.