In this section, we will look at several attempts to, once a hyperbolic system has been obtained for the evolution variables, extend it into a bigger well posed system where the lapse-shift pair, or more generally the gauge variables of a particular system, can be determined by some prescription which in general depends on the dynamical variables. The bigger system does not need to be hyperbolic, it only needs to be well posed, and so for instance it can be mixed elliptic-hyperbolic.

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