  A K\"ahler metric is called central if the determinant of its Ricci
endomorphism is constant. For the case in which this constant is zero, we study
on $4$-manifolds the existence of complete metrics of this type which are
cohomogeneity one for three unimodular $3$-dimensional Lie groups: $SU(2)$, the
group of Euclidean plane motions $E(2)$ and a quotient by a discrete subgroup
of the Heisenberg group $\mathrm{nil}_3$. We obtain a complete classification
for $SU(2)$, and some existence results for the other two groups, in terms of
specific solutions of an associated ODE system.
