A solution to the Cauchy dual subnormality problem for a cyclic analytic $2$-isometry with defect operator of rank two

The Cauchy dual subnormality problem (for short, CDSP) asks whether the Cauchy dual of a $2$-isometry is subnormal. In this article, we prove that if $\mu$ is a sum of unit point mass measures at two non-antipodal points on the unit circle, then the Cauchy dual $M_z^{\mathrm{\prime }}$ of the multiplication operator $M_z$ on the Dirichlet-type space $D(\mu )$ is not subnormal. If the points are antipodal then the subnormality of the said operator has been already established in the literature. Thus, we have a complete solution to CDSP in this case.