  We study a vector field of R^3 equivariant under the D_2 symmetry group,
called "the D_2 field" in the literature. We construct the complete list of
Darboux polynomials for it, solving the partial differential equation defining
them. We also use these polynomials to comment on its global qualitative
behaviour. This is meant to be a first step towards the comparison of vector
fields based on the module generated by their Darboux polynomials.
