Vector potential normal form classification for completely integrable solenoidal nilpotent singularities

We introduce an sl 2 -invariant family of polynomial vector fields with an irreducible nilpotent singularity. In this paper, we are concerned with characterization and normal form classification of these vector fields. We show that the family is a Lie subalgebra and each vector field from this family is volume-preserving, completely integrable, and rotational. All such vector fields share a common quadratic invariant. We provide a Poisson structure for the Lie subalgebra 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 which can be found in applications: 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.