  We consider a class of semidirect products $G = \mathbb{R}^n \rtimes H$, with
$H$ a suitably chosen abelian matrix group. The choice of $H$ ensures that
there is a wavelet inversion formula, and we are looking for criteria to decide
under which conditions there exists a wavelet such that the associated
reproducing kernel is integrable.
  It is well-known that the existence of integrable wavelet coefficients is
related to the question whether the unitary dual of $G$ contains open compact
sets. Our main general result reduces the latter problem to that of identifying
compact open sets in the quotient space of all orbits of maximal dimension
under the dual action of $H$ on $\mathbb{R}^n$. This result is applied to study
integrability for certain families of dilation groups; in particular, we give a
characterization valid for connected abelian matrix groups acting in dimension
three.
