  The Einstein-Aether theory is an alternative theory of gravity in which the
spacetime metric is supplemented by a long-range timelike vector field (the
"aether" field). Here, for the first time, we apply the full formalism of
post-Minkowskian theory and of the Direct Integration of the Relaxed Einstein
Equations (DIRE), to this theory of gravity, with the goal of deriving
equations of motion and gravitational waveforms for orbiting compact bodies to
high orders in a post-Newtonian expansion. Because the aether field is
constrained to have unit norm, a naive application of post-Minkowskian theory
leads to contributions to the effective energy momentum tensor that are {\em
linear} in the perturbative fields. We show that a suitable redefinition of
fields using an array of "superpotentials" can eliminate such linear terms to
any desired post-Newtonian order, resulting in flat spacetime wave equations
for all fields, with sources consisting of matter terms and terms quadratic and
higher in the fields. As an initial application of this new method, and as a
foundation for obtaining the equations of motion for compact binaries, we
obtain explicit solutions of the relaxed equations sufficient to obtain the
metric in the near zone through 2.5 post-Newtonian order, or $O[(v/c)^5]$
beyond the Newtonian approximation.
