$q, \, \alpha, \, \beta, \, \Gamma \equiv  \Gamma_0\,^1\,_0 + i\, \Gamma_0\,^2\,_0
\in C^{\infty}({\cal T}, \mathbb{C})$ which are of spin-weight $-2, -4, 0, 1$\
respectively and so that $$\left[
\begin{array}{cc}
Re(\bar{\alpha}\,\beta) - \frac{1}{2}(1 - |\alpha|^2 - |\beta|^2)
& Im(\bar{\alpha}\,\beta) \\
Im(\bar{\alpha}\,\beta) & 
- Re(\bar{\alpha}\,\beta) - \frac{1}{2}(1 - |\alpha|^2 - |\beta|^2)
\\
\end{array}
\right] \le 0,$$ in the sense of quadratic forms.

The following boundary condition is required on ${\cal T}$: $$\label{bdry-cond-*}
q = - \Psi_4 + \alpha\,\Psi_0 +
\beta\,\bar{\Psi}_0,$$ where $$\quad\mbox{with}\quad
\Psi_0 = C_{\mu \nu \lambda \rho}
\,l^{\mu}\,m^{\nu}\,l^{\lambda}\,m^{\rho},\,\,\,\,\,\,\,\,
\Psi_4 = C_{\mu \nu \lambda \rho}
\,\bar{m}^{\mu}\,k^{\nu}\,\bar{m}^{\lambda}\,k^{\rho}.$$

Properties of the setting and the main problem
----------------------------------------------

If the gauge source functions and the functions $\alpha$, $\beta$ have been prescribed, the symmetric hyperbolic reduced system and the Cauchy data allows us to determine for all unknowns their time derivatives at of any order on ${\cal S}$. The boundary condition (\[bdry-cond-\*\]) then gives us restrictions on the function $q$ on $\Sigma$. If $q$ is chosen accordingly, the consistency conditions will be satisfied.

Because the gauge conditions of the present setting are explicit, an analogue of (\[gamma=0-on-T\]) is not required. Instead of controlling the gauge condition one has to control constraints. This task more or 