  We prove local existence of solutions to the Einstein--null dust system under
polarized $\mathbb U(1)$ symmetry in an elliptic gauge. Using in particular the
previous work of the first author on the constraint equations, we show that one
can identify freely prescribable data, solve the constraints equations, and
construct a unique local in time solution in an elliptic gauge. Our main
motivation for this work, in addition to merely constructing solutions in an
elliptic gauge, is to provide a setup for our companion paper in which we study
high frequency backreaction for the Einstein equations. In that work, the
elliptic gauge we consider here plays a crucial role to handle high frequency
terms in the equations. The main technical difficulty in the present paper, in
view of the application in our companion paper, is that we need to build a
framework consistent with the solution being high frequency, and therefore
having large higher order norms. This difficulty is handled by exploiting a
reductive structure in the system of equations.
