  The conformal method has been effective for parametrizing solutions to the
Einstein constraint equations on closed 3-manifolds. However, it is still not
well-understood; for example, existence of solutions to the conformal equations
for zero or negative Yamabe metrics is still unknown without the so-called
``CMC'' or ``near-CMC'' assumptions. The first existence results without such
assumptions, termed the ``far-from-CMC'' case, were obtained by Holst, Nagy,
and Tsogtgerel in 2008 for positive Yamabe metrics. However, their results are
based on topological arguments, and as a result solution uniqueness is not
known. Indeed, Maxwell gave evidence in 2011 that far-from-CMC solutions are
not unique in certain cases. In this article, we provide further insight by
establishing a type of alternative theorem for general far-from-CMC solutions.
For a given manifold M that admits a metric of positive scalar curvature and
scalar flat metric g(0) with no conformal Killing fields, we first prove
existence of an analytic, one-parameter family of metrics g(z) through g(0)
such that R(g(z)) = z. Using this family of metrics and given data
(tau,sigma,rho,j), we form a one-parameter family of operators F((phi,w),z)
whose zeros satisfy the conformal equations. Applying Liapnuov-Schmidt
reduction, we determine an analytic solution curve for F((phi,w),z) = 0 through
a critical point where the linearization of F((phi,w),z) vanishes. The
regularity of this curve, the definition of F((phi,w),z), and the earlier
far-from-CMC results of Holst et al. allow us to then prove the following
alternative theorem for far-from-CMC solutions: either (1) there exists a z_1
>0 such that (positive Yamabe) solutions to the z_1-parameterized conformal
equations are non-unique; or (2) there exists z_2 < 0 such that (negative
Yamabe) solutions to the z_2-parameterized conformal equations exist.
