Doubly separable spacetimes and symmetry constraints on their self-gravitating matter content

A popular approach to constructing exact stationary and axisymmetric nonvacuum solutions in general relativity has been to use solution-generating techniques. Here we revisit a recent variant of the Newman-Janis-Azreg-A"{\i}nou algorithm---restricted to asymptotically flat spacetimes---and demonstrate that this method exclusively generates Konoplya-Stuchl\'{\i}k-Zhidenko spacetimes. Therefore, the equations for geodesic motion and scalar-wave propagation are both separable. We call these ``doubly separable'' spacetimes. Of these, we identify a ``degenerate'' subclass that might admit a separable Dirac equation by explicitly obtaining the Killing-Yano tensor. While the degenerate subclass is Petrov type D, the general doubly separable spacetimes are of type I. The high degree of symmetry in these spacetimes suggests that the self-gravitating matter must also be in specialized field configurations. For this reason, we investigate whether these spacetimes can even be sourced by arbitrary types of matter. We show that doubly separable spacetimes cannot be sourced by massless real scalar fields or perfect fluids, and that electromagnetic fields lead only to the Kerr-Newman family. Notably, this rules out the correct spinning counterpart of the Janis-Newman-Winicour naked singularity spacetime, which contains a scalar field, as a member of this metric class. While the algorithm generates spacetimes with rich symmetry structures, valuable for studying phenomena like black hole shadows and quasinormal modes, our results highlight the need for caution when using it to construct physically consistent solutions with prespecified matter content.