On the existence of limit cycles and invariant surfaces of sewing piecewise linear differential systems on ${\mathbb{R}}^3$

We consider a class of discontinuous piecewise linear differential systems in ${\mathbb{R}}^3$ with two pieces separated by a plane. In this class we show that there exist differential systems having: a unique limit cycle, a unique one-parameter family of periodic orbits, scrolls, invariant cylinders foliated by orbits which can be periodic or no.