  The Kasner metrics are among the simplest solutions of the vacuum Einstein
equations, and we use them here to examine the conformal method of finding
solutions of the Einstein constraint equations. After describing the conformal
method's construction of constant mean curvature (CMC) slices of Kasner
spacetimes, we turn our attention to non-CMC slices of the smaller family of
flat Kasner spacetimes. In this restricted setting we obtain a full description
of the construction of certain $U^{n-1}$ symmetric slices, even in the
far-from-CMC regime. Among the conformal data sets generating these slices we
find that most data sets construct a single flat Kasner spacetime, but that
there are also far-from-CMC data sets that construct one-parameter families of
slices. Although these non-CMC families are analogues of well-known CMC
one-parameter families, they differ in important ways. Most significantly,
unlike the CMC case, the condition signaling the appearance of these non-CMC
families is not naturally detected from the conformal data set itself. In light
of this difficulty, we propose modifications of the conformal method that
involve a conformally transforming mean curvature.
