  In this paper we identify a new family of black holes and solitons that lead
to the exact integration of scalar probes, even in the presence of a
non-minimal coupling with the Ricci scalar which has a non-trivial profile. The
backgrounds are planar and spherical black holes as well as solitons of
$SU\left( 2\right) \times SU\left( 2\right) $ $\mathcal{N}=4$ gauged
supergravity in four dimensions. On these geometries, we compute the spectrum
of (quasi-)normal modes for the non-minimally coupled scalar field. We find
that the equation for the radial dependence can be integrated in terms of
hypergeometric functions leading to an exact expression for the frequencies.
The solutions do not asymptote to a constant curvature spacetime, nevertheless
the asymptotic region acquires an extra conformal Killing vector. For the black
hole, the scalar probe is purely ingoing at the horizon, and requiring that the
solutions lead to an extremum of the action principle we impose a Dirichlet
boundary condition at infinity. Surprisingly, the quasinormal modes do not
depend on the radius of the black hole, therefore this family of geometries can
be interpreted as isospectral in what regards to the wave operator
non-minimally coupled to the Ricci scalar. We find both purely damped modes, as
well as exponentially growing unstable modes depending on the values of the
non-minimal coupling parameter. For the solitons we show that the same
integrability property is achieved separately in a non-supersymmetric solutions
as well as for the supersymmetric one. Imposing regularity at the origin and a
well defined extremum for the action principle we obtain the spectra that can
also lead to purely oscillatory modes as well as to unstable scalar probes,
depending on the values of the non-minimal coupling.
