  Standard methods in non-linear analysis are used to show that there exists a
parabolic branching of solutions of the Lichnerowicz-York equation with an
unscaled source. We also apply these methods to the extended conformal thin
sandwich formulation and show that if the linearised system develops a kernel
solution for sufficiently large initial data then we obtain parabolic solution
curves for the conformal factor, lapse and shift identical to those found
numerically by Pfeiffer and York. The implications of these results for
constrained evolutions are discussed.
