  In this paper, we deal with the solenoidal conservative Lie algebra
associated to the classical normal form of Hopf-zero singular system. We
concentrate on the study of some representations and $\mathbb{Z}_2$-equivariant
normal form for such singular differential equations. First, we list some of
the representations that this Lie algebra admits. The vector fields from this
Lie algebra could be expressed by the set of ordinary differential equations
where the first two of them are in the canonical form of a one-degree of
freedom Hamiltonian system and the third one depends upon the first two
variables. This representation is governed by the associated Poisson algebra to
one sub-family of this Lie algebra. Euler's form, vector potential, and Clebsch
representation are other representations of this Lie algebra that we list here.
We also study the non-potential property of vector fields with Hopf-zero
singularity from this Lie algebra. Finally, we examine the unique normal form
with non-zero cubic terms of this family in the presence of the symmetry group
$\mathbb{Z}_2$ . The theoretical results of normal form theory are illustrated
with the modified Chua's oscillator.
