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Our start point is a 3D piecewise smooth vector field defined in two zonesand 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 eachorthogonal section of these cylinders is filled by centers. First we prove thatthe normal form considered represents a whole class of piecewise smooth vectorfields. After we perturb the initial model in order to obtain exactly$\mathcal{L}$ invariant planes containing centers. A second perturbation of theinitial model also is considered in order to obtain exactly $k$ isolatedcylinders filled by periodic orbits. Finally, joining the two previousbifurcations we are able to exhibit a model, preserving the symmetry relativeto the fold curve, and having exactly $k.\mathcal{L}$ limit cycles.

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