Normal forms for Hopf-Zero singularities with nonconservative nonlinear part

In this paper we are concerned with the simplest normal form computation of the systemsequation(0.1)x˙=2xf(x,y2+z2),y˙=z+yf(x,y2+z2),z˙=−y+zf(x,y2+z2), where f is a formal function with real coefficients and without any constant term. These are the classical normal forms of a larger family of systems 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 nonzero 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 is 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.