2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/183071We consider 5D Einstein-Maxwell (EM) gravity in spacetimes with three commuting Killing vectors: one timelike and two spacelike Killing vectors one of them being hypersurface-orthogonal. Assuming a special ansatz for the Maxwell field we show that the 2-dimensional reduced EM equations are completely integrable by deriving a Lax-pair presentation. We also develop a solution generating method for explicit construction of exact EM solutions with considered symmetries. We also derive explicitly a new rotating six parametric 5D EM solution which includes the dipole black ring solution as a particular case.LaTex, 13 pages; v2 some typos corrected, references added;v3 more typos corrected, more references added, some explanations expanded, PRDHigh Energy Physics - TheoryGeneral Relativity and Quantum Cosmologytext