  A motivation for studying the following problems comes from applications to
Biology; see \cite{cifuentes20233d}. In the $3$-dimensional Euclidean space
${\bf{E}}^3$, fix six pairwise distinct points \begin{equation*} \label{eqA}
\begin{array}{ccc} A=(a_1,a_2,a_3), & B=(b_1,b_2,b_3), & C=(c_1,c_2,c_3), \\
D=(d_1,d_2,d_3), & E=(e_1,e_2,e_3), & F=(f_1,f_2,f_3) \end{array}
\end{equation*} together with two further points $X^*=(x_1^*,x_2^*,x_3^*)$ and
$Y^*=(y_1^*,y_2^*,y_3^*)$ in $\mathbf{E}^3$. We aim to show that System $(*)$
consisting of the following six equations in the unknowns $X=(x_1,x_2,x_3)$ and
$Y=(y_1,y_2,y_3)$ \begin{equation} \label{egy} \frac{1}{\|X-T\|^2}
+\frac{1}{\|Y-T\|^2}=\frac{1}{\|X^*-T\|^2} +\frac{1}{\|Y^*-T\|^2}, \quad
T\in\{A,B,\ldots,F\}. \end{equation} has only finitely many solutions provided
that both of the following two conditions are satisfied: (i) no four of the
fixed points $A,B,C,D,E,F$ are coplanar; (ii) no four of the six spheres of
center $T$ and radius $1/\sqrt{k_T}$ with \begin{equation} \label{kxy}
k_T=\frac{1}{\|X^*-T\|^2} +\frac{1}{\|Y^*-T\|^2} \end{equation} share a common
point in ${\bf{E}}^3$. Furthermore, we exhibit configurations $ABCDEFX^*Y^*$,
showing that (i) is also necessary. This result is an improvement on
\cite[Theorem 1]{cifuentes20233d} where the finiteness of solutions of System
$(*)$ is only ensured for sufficiently generic choices of the points
$A,B,\ldots,F,X^*,Y^*.$ We also show if System $(*)$ has finitely many
solutions and System $(*)$ extended with $T\in\{A,B,\ldots,F,G\}$ has some
solutions other than $(X^*,Y^*)$ and $(Y^*,X^*)$ then $G$ lies on an explicitly
given affine variety $W\subsetneqq \mathbb{R}^3$ only depending on
$\{A,B,\ldots,F\}$. This result proves the \cite[Conjecture
1]{cifuentes20233d}.
