  We 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.
