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

Compact binary systems in Einstein-Aether gravity: Direct integration of the relaxed field equations to 2.5 post-Newtonian order