the parameter space of the ${\mathcal{G}_6}$ manifold (see [@GR02]). An algebraic solution of these equations for $(x,y,z)$ in terms of $(|a|,|b|,|c|)$, or vice versa, would in general yield higher degree (decoupled) equations for each variable, and thus are not so illuminating. Particular solutions can be obtained by (i) assuming specific values for the parameters $(|a|,|b|,|c|)$, and then calculating numerically the position of $C_0$, or (ii) assuming some particular position for $C_0$, and then calculating the parameters $(|a|,|b|,|c|)$. This second method does not follow the strategy of determining the parameters of the manifold using two more triples of clusters of galaxies (see Section \[Decide\]), thus it will not be pursued here. The next section presents examples of application of the first method.

Finally, let us examine the second configuration which is simpler. One has $\delta_{\beta}(p)=d_1$, $\delta_{\mu}(p)=d_2$ and $\cos(\beta,\mu) = \cos\theta$, thus yielding the equations $$\begin{aligned}
       4x^2 + (2z - |c|)^2 & = & d_1^2 - |b|^2 \nonumber \\
\label{EqSecondG6}
       4x^2 + (2z + |c|)^2 & = & d_2^2 - |b|^2 \\
4x^2 + 4z^2 + |b|^2 -|c|^2 & = & d_1d_2\cos\theta \; . \nonumber \end{aligned}$$ These equations can be partially solved giving $$\begin{aligned}
        z & = & \frac{1}{8|c|} \, (d_2^2 - 