Generality of the separable stationary and axisymmetric scalar profile

Determine whether a separable stationary and axisymmetric scalar field profile Φ(r,θ) of the form Φ(r,θ)=Φr(r)+Φθ(θ) is the most general real, massless scalar solution compatible with the stationary and axisymmetric Einstein–Scalar metrics constructed by rescaling only the non-Killing (r,θ) sector of a rotating vacuum seed (such as Kerr, Kerr–Newman–NUT, or Myers–Perry), or identify more general non-separable scalar configurations if they exist.

Background

The paper develops stationary and axisymmetric solutions of Einstein gravity minimally coupled to a free scalar by modifying only the non-Killing (r,θ) sector of known rotating vacuum seeds, yielding explicit four- and five-dimensional metrics. In several constructions (e.g., Kerr–Newman–NUT and five-dimensional Myers–Perry–like cases), the scalar field is taken to have a separable dependence on r and θ, with two independent scalar charges.

Although explicit solutions are obtained with this separable ansatz, the authors note they have not established whether separability exhausts the space of stationary and axisymmetric scalar solutions compatible with their metric ansatz. Clarifying this would amount to a classification of allowed scalar profiles under the assumed symmetry and metric structure.