Solutions to the Einstein's equations approximately describe physical objects like stars, what happens when one attempts to make more accurate descriptions?
The Kerr metric represents a massive spinning body (such as any physical star), the static limit is the expected Schwarzschild metric, i.e. a massive spherical body. Both of these metrics are exterior metrics, thus an inner matching to an appropriate energy momentum tensor must exist. Nevertheless, finding an interior matching for the Kerr metric has proven to be quite a challenge.
One possible solution is to consider metrics that are more general than the Kerr metric by relaxing the geometry -- deviations from perfect spheres -- such that one has more degrees of freedom to do the matching. However, it does not matter what modifications you do to these approximate metrics they must be a limiting case of the very successful Kerr metric at a certain limit.
For more information please visit: https://arxiv.org/abs/1405.2899, https://arxiv.org/pdf/1405.1776.pdf and https://arxiv.org/abs/1609.00102