  The conformal formulation provides a method for constructing and
parametrizing solutions of the Einstein constraint equations by mapping freely
chosen sets of conformal data to solutions, provided a certain set of coupled,
elliptic determined PDEs (whose expression depends on the chosen conformal
data) admit a unique solution. For constant mean curvature (CMC) data, it is
known in almost all cases which sets of conformal data allow these PDEs to have
solutions, and which do not. For non CMC data, much less is known. Here we
exhibit the first class of non CMC data for which we can prove that no
solutions exist.
