Birth of isolated nested cylinders and limit cycles in 3D piecewise smooth vector fields with symmetry

Our start point is a 3D piecewise smooth vector field defined in two zones and presenting a shared fold curve for the two smooth vector fields considered. Moreover, these smooth vector fields are symmetric relative to the fold curve, giving raise to a continuum of nested topological cylinders such that each orthogonal section of these cylinders is filled by centers. First we prove that the normal form considered represents a whole class of piecewise smooth vector fields. After we perturb the initial model in order to obtain exactly $\mathcal{L}$ invariant planes containing centers. A second perturbation of the initial model also is considered in order to obtain exactly $k$ isolated cylinders filled by periodic orbits. Finally, joining the two previous bifurcations we are able to exhibit a model, preserving the symmetry relative to the fold curve, and having exactly $k.\mathcal{L}$ limit cycles.