  The R\"ossler System is characterized by a three-parameter family of
quadratic 3D vector fields. There exist two one-parameter families of R\"ossler
Systems exhibiting a zero-Hopf equilibrium. For R\"ossler Systems near to one
of these families, we provide generic conditions ensuring the existence of a
torus bifurcation. In this case, the torus surrounds a periodic solution that
bifurcates from the zero-Hopf equilibrium. For R\"ossler Systems near to the
other family, we provide generic conditions for the existence of a periodic
solution bifurcating from the zero-Hopf equilibrium. This improves currently
known results regarding periodic solutions for such a family. In addition, the
stability properties of the periodic solutions and invariant torus are
analysed.
