  The Lie algebra generated by Hopf-zero classical normal forms is decomposed
into two versal Lie subalgebras. Some dynamical properties for each subalgebra
are described; one is the set of all volume-preserving conservative systems
while the other is the maximal Lie algebra of nonconservative systems. This
introduces a unique conservative--nonconservative decomposition for the normal
form systems. There exists a Lie--subalgebra that is Lie-isomorphic to a large
family of vector fields with Bogdanov--Takens singularity. This gives rise to a
conclusion that the local dynamics of formal Hopf-zero singularities is
well-understood by the study of Bogdanov--Takens singularities. Despite this,
the normal form computation of Bogdanov-Takens and Hopf-zero singularities are
independent. Thus, by assuming a quadratic non-zero condition, complete results
on the simplest Hopf-zero normal forms are obtained in terms of the
conservative--nonconservative decomposition. Some practical formulas are
derived and the results implemented using Maple. The method has been applied on
the R\"ossler and Kuramoto--Sivashinsky equations to demonstrate the
applicability of our results.
