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.