  We consider a particular truncation of the generalized Proca field theory in
four dimensions for which we construct a static and axisymmetric rotating black
hole "stealth solutions", namely solutions with (Anti) de Sitter or Kerr metric
but non-trivial vector field. The static configuration is promoted to a
backreacting black hole with asymptotic (Anti) de Sitter behavior by turning on
a nonlinear electrodynamic source given as a fixed power of the Maxwell
invariant. Finally we extend our solutions to arbitrary dimensions.
