  In this paper some piecewise smooth perturbations of a three-dimensional
differential system are considered. The existence of invariant manifolds filled
by periodic orbits is obtained after suitable small perturbations of the
original differential system. These manifolds emerge from a continuum of
cylinders of $\mathbb{R}^3$ which does exist for the piecewise smooth
differential systems after a rotation of some planar algebraic polynomial
curves. The main tool used in order to obtain the results is the averaging
theory for piecewise smooth differential systems.
