  Finding the maximum cardinality of a $2$-distance set in Euclidean space is a
classical problem in geometry. Lison\v{e}k in 1997 constructed a maximum
$2$-distance set in $\mathbb R^8$ with $45$ points. That $2$-distance set
constructed by Lison\v{e}k has a distinguished structure of a coherent
configuration of type $(2,2;3)$ and is embedded in two concentric spheres in
$\mathbb R^8$. In this paper we study whether there exists any other similar
embedding of a coherent configuration of type $(2,2;3)$ as a $2$-distance set
in $\mathbb R^n$, without assuming any restriction on the size of the set. We
prove that there exists no such example other than that of Lison\v{e}k. The key
ideas of our proof are as follows: (i) study the geometry of the embedding of
the coherent configuration in Euclidean spaces and to drive diophantine
equations coming from this embedding. (ii) solve diophantine equations with
certain additional conditions of integrality of some parameters of the
combinatorial structure by using the method of auxiliary equations.
