\Gamma_0\,^A\,_0\,e_A, \quad
D_{e_0}\,e_A = - g_{AB}\,\Gamma_0\,^B\,_0\,e_0, \quad A, B = 1, 2,$$ where the summation convention applies to both groups of indices.

As discussed in [@friedrich:nagy], the three functions $\Gamma_0\,^A\,_B$, $\Gamma_0\,^A\,_0$ play the role of gauge source functions on ${\cal M}$ (cf. [@friedrich:2hyp; @red]) and can be chosen arbitrarily. The first of these functions has been disposed of here in a convenient way, choosing the other two functions represents the usual gauge problem in the interior of ${\cal M}$ but it is delicate task on the boundary. The function $\chi$, which can also be prescribed, plays the role of a gauge source function in the interior of ${\cal M}$ and the role of a boundary datum on ${\cal T}$.

With this gauge it is easy to extracted from the complete, overdetermined system (\[first-structural\]), (\[second-structural\]), (\[bianchi\]) a ‘reduced system’ for the unknowns (that are not gauge source functions) which is symmetric hyperbolic and for which well-posed initial boundary value problems can be formulated. But in general the resulting evolution will not preserve the constraints so that not all of equations (\[first-structural\]), (\[second-structural\]), (\[bianchi\]) will be satisfied.

The main theorem of [@friedrich:nagy] shows, however, that there do exist ‘reduced systems’ for which 