Generic Bifurcations of Symmetric Two-zone Piecewise Vector Fields on R³

Inspired by Thom–Smale’s program on the qualitative classification of dynamical systems, we develop a structural framework for the local analysis of symmetric two-zone piecewise smooth vector fields on R³ R 3 . The framework is based on a natural correspondence with vector fields on manifolds with boundary, allowing local qualitative properties to be transferred to the symmetric piecewise setting. As applications, we derive complete classifications of generic codimension-zero and codimension-one R -reversible and R -equivariant symmetric singularities.