  In this article, we are interested in the Einstein vacuum equations on a
Lorentzian manifold displaying $\mathbb{U}(1)$ symmetry. We identify some
freely prescribable initial data, solve the constraint equations and prove the
existence of a unique and local in time solution at the $H^3$ level. In
addition, we prove a blow-up criterium at the $H^2$ level. Our main objective
is to provide a framework adapted to the study of high-frequency solutions to
the Einstein vacuum equations done in a forthcoming paper by Huneau and Luk. As
a consequence we work in an elliptic gauge, particularly adapted to the
handling of high-frequency solutions, which have large high-order norms.
