  We introduce a sl_2-invariant family of nonlinear vector fields with a
non-semisimple triple zero singularity. In this paper we are concerned with
characterization and normal form classification of these vector fields. We show
that the family constitutes a Lie algebra structure and each vector field from
this family is solenoidal, completely integrable and rotational. All such
vector fields share a common quadratic invariant. We provide a Poisson
structure for the Lie algebra from which the second invariant for each vector
field can be readily derived. We show that each vector field from this family
can be uniquely characterized by two alternative representations, one uses a
vector potential while the other uses two functionally independent Clebsch
potentials. Our normal form results are designed to preserve these structures
and representations. The results are implemented in Maple in order to compute
vector potential and the Clebsch potential normal forms of a given vector field
from this family. Some practical normal form coefficient formulas for degrees
of up to four are presented.
