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We consider 5D Einstein-Maxwell (EM) gravity in spacetimes with threecommuting Killing vectors: one timelike and two spacelike Killing vectors oneof them being hypersurface-orthogonal. Assuming a special ansatz for theMaxwell field we show that the 2-dimensional reduced EM equations arecompletely integrable by deriving a Lax-pair presentation. We also develop asolution generating method for explicit construction of exact EM solutions withconsidered symmetries. We also derive explicitly a new rotating six parametric5D EM solution which includes the dipole black ring solution as a particularcase.Comment: LaTex, 13 pages; v2 some typos corrected, references added;v3 more  typos corrected, more references added, some explanations expanded, PR

Completely integrable sector in 5D Einstein-Maxwell gravity and derivation of the dipole black ring solutions