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In this paper, we deal with the solenoidal conservative Lie algebraassociated to the classical normal form of Hopf-zero singular system. Weconcentrate on the study of some representations and $\mathbb{Z}_2$-equivariantnormal form for such singular differential equations. First, we list some ofthe representations that this Lie algebra admits. The vector fields from thisLie algebra could be expressed by the set of ordinary differential equationswhere the first two of them are in the canonical form of a one-degree offreedom Hamiltonian system and the third one depends upon the first twovariables. This representation is governed by the associated Poisson algebra toone sub-family of this Lie algebra. Euler's form, vector potential, and Clebschrepresentation are other representations of this Lie algebra that we list here.We also study the non-potential property of vector fields with Hopf-zerosingularity from this Lie algebra. Finally, we examine the unique normal formwith 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 illustratedwith the modified Chua's oscillator.

On the representations and $\mathbb{Z}_2$-equivariant normal form for solenoidal Hopf-zero singularities