  The drift method, introduced by the second author, provides a new formulation
of the Einstein constraint equations, either in vacuum or with matter fields.
The natural of the geometry underlying this method compensates for its slightly
greater analytic complexity over, say, the conformal or conformal thin sandwich
methods. We review this theory here and apply it to the study of solutions of
the constraint equations with non-constant mean curvature. We show that this
method reproduces previously known existence results obtained by other methods,
and does better in one important regard. Namely, it can be applied even when
the underlying metric admits conformal Killing (but not true Killing) vector
fields. We also prove that the absence of true Killing fields holds
generically.
