<p>The literature is replete with diverse research endeavors on the Einstein–Maxwell equations (EMEs). The EMEs are the coupled field equations that describe how gravitation (spacetime curvature) and electromagnetism interact within the framework of general relativity. They combine Einstein’s field equations with Maxwell’s equations.The present study, however, is concerned with the derivation of similarity solutions for axially-symmetric fields. To achieve this, we introduce similarity transformations and implement the extended unified method, a robust approach to tackling complex field equations. Our objective is to derive solutions for several key parameters: mass, rotation, electrostatic, and electromagnetic potentials. These parameters are intrinsically linked with energy, making their understanding crucial to the study of these fields. The results reveal that the electrostatic and electromagnetic potentials manifest as solitary waves. These waves exhibit forward or backward motion, albeit with varying magnitudes. Furthermore, the amplitudes of these waves display a range of phenomena. These include the kink solitons packets self-trap due to scalar-induced nonlinearity, interactions between lumps and holes (termed lump-tunneling interactions), and wave fusion and fission. Furthermore, this work provides valuable insights into the behavior of axially-symmetric fields, contributing to our understanding of gravitational and electromagnetic fields within the framework of the Einstein–Maxwell equations.</p>