  We study the appearance of multiple solutions to certain decompositions of
Einstein's constraint equations. Pfeiffer and York recently reported the
existence of two branches of solutions for identical background data in the
extended conformal thin-sandwich decomposition. We show that the Hamiltonian
constraint alone, when expressed in a certain way, admits two branches of
solutions with properties very similar to those found by Pfeiffer and York. We
construct these two branches analytically for a constant-density star in
spherical symmetry, but argue that this behavior is more general. In the case
of the Hamiltonian constraint this non-uniqueness is well known to be related
to the sign of one particular term, and we argue that the extended conformal
thin-sandwich equations contain a similar term that causes the breakdown of
uniqueness.

General Relativity and Quantum Cosmology