  In this work we construct and analyse non-perturbative stationary and
axially-symmetric black hole solutions in General Relativity coupled to an
electromagnetic and an axion field. The axion field is coupled to the
electromagnetic field, which leads to hairy solutions in the presence of an
electric charge and rotation. We investigate the existence and characteristics
of these solutions for different values of the spin, charge and coupling
constant. Our analysis shows the presence of violations of the Kerr-Newman
bound, solutions with large positive and negative values of the gyromagnetic
ratio, and the existence of multiple branches of solutions with distinct
properties, demonstrating that black hole uniqueness does not hold in this
scenario. The code used in this study is publicly available, providing a
valuable tool for further research on this model.
