Normal Form of Hopf-Zero Reversible Solenoidal Singularities

In this paper we study a family of solenoidal Hopf-zero singularities with one type of reversibility. Our goal is to introduce this family and classify the simplest normal form of this singular vector field. The simplest normal form for this family is provided by considering a nonzero condition on their quadratic terms. The consequences are applied to KuramotoSivashinsky equations to demonstrate the applicability of our theoretical results. This is a part of our results in studying the simplest normal form and bifurcation analysis of Hopf-zero singular system with reversibility.