  In this paper we are concerned with the simplest normal form computation of a
family of Hopf-zero vector fields without a first integral. This family of
vector fields are the classical normal forms of a larger family of vector
fields with Hopf-Zero singularity. Indeed, these are defined such that this
family would be a Lie subalgebra for the space of all classical normal form
vector fields with Hopf-Zero singularity. The simplest normal forms and
simplest orbital normal forms of this family with non-zero quadratic part are
computed. We also obtain the simplest parametric normal form of any
non-degenerate perturbation of this family within the Lie subalgebra. The
symmetry group of the simplest normal forms are also discussed. This is a part
of our results in decomposing the normal forms of Hopf-Zero singular systems
into systems with a first integral and nonconservative systems.
