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We extend the study of the vacuum Einstein constraint equations on manifoldswith ends of cylindrical type initiated by Chru\'sciel and Mazzeo by finding aclass of solutions to the fully coupled system on such manifolds. We show thatgiven a Yamabe positive metric g, which is conformally asymptoticallycylindrical on each end, and a 2-tensor K such that (tr K)^2 is bounded belowaway from zero and asymptotically constant, then we may find an initial dataset (g',K') such that g' lies in the conformal class of g.

A far-from-CMC existence result for the constraint equations on manifolds with ends of cylindrical type