SIMPLEST NORMAL FORMS AND A CERTAIN FAMILY OF HOPF-ZERO VECTOR FIELDS WITH FIRST INTEGRAL

The classical normal form space of the Hopf-Zero singularities is decomposed into three families of dynamically meaningful and important vector fields, where they each generate a Lie subalgebra for the whole space; one has a first integral, the other is quasi- Eulerian, and the last one is associated with phase components of vector fields in cylindrical coordinates. The focus here is on the first family of vector fields. We use an sl2 orbit structure to establish a natural sl2-representation and then, use this to obtain their simplest normal forms.