  The local normal zeta functions of a finitely generated, torsion-free
nilpotent group $G$ of class 2 depend on the geometry of the Pfaffian
hypersurface associated to the bilinear form induced by taking commutators in
$G$.
  The smallest examples of zeta functions which are not finitely uniform arise
from groups whose associated Pfaffian hypersurfaces are plane curves. In this
paper we study groups whose Pfaffians define singular curves, illustrating that
the local normal zeta functions may indeed invoke all the degeneracy loci of
the Pfaffian.
