  A practical method is described for computing the unique generator of the
algebra of first integrals associated with a large class of Hopf-zero
singularity. The set of all volume-preserving classical normal forms of this
singularity is introduced via a Lie algebra description. This is a maximal
vector space of classical normal forms with first integral; this is whence our
approach works. Systems with a non-zero condition on their quadratic parts are
considered. The algebra of all first integrals for any such system has a unique
(modulo scalar multiplication) generator. The infinite level volume-preserving
parametric normal forms of any non-degenerate perturbation within the Lie
algebra of any such system is computed, where it can have rich dynamics. The
associated unique generator of the algebra of first integrals are derived. The
symmetry group of the infinite level normal forms are also discussed. Some
necessary formulas are derived and applied to appropriately modified
R\"{o}ssler and generalized Kuramoto--Sivashinsky equations to demonstrate the
applicability of our theoretical results. An approach (introduced by Iooss and
Lombardi) is applied to find an optimal truncation for the first level normal
forms of these examples with exponentially small remainders. The numerically
suggested radius of convergence (for the first integral) associated with a
hypernormalization step is discussed for the truncated first level normal forms
of the examples. This is achieved by an efficient implementation of the results
using Maple.
