  The Cauchy dual subnormality problem (for short, CDSP) asks whether the
Cauchy dual of a $2$-isometry is subnormal. In this paper, we address this
problem for cyclic $2$-isometries. In view of some recent developments in
operator theory on function spaces (see \cite{AM, LGR}), one may recast CDSP as
the problem of subnormality of the Cauchy dual $\mathscr M'_z$ of the
multiplication operator $\mathscr M_z$ acting on a de Branges-Rovnyak space
$\mathcal H(B),$ where $B$ is a vector-valued rational function. The main
result of this paper characterizes the subnormality of $\mathscr M'_z$ on
$\mathcal H(B)$ provided $B$ is a vector-valued rational function with simple
poles. As an application, we provide affirmative solution to CDSP for the
Dirichlet-type spaces $\mathscr D(\mu)$ associated with measures $\mu$
supported on two antipodal points of the unit circle.
