  The main goal of the paper is to establish a sufficient condition for a
two-valenced association scheme to be schurian and separable. To this end, an
analog of the Desargues theorem is introduced for a noncommutative geometry
defined by the scheme in question. It turns out that if the geometry has enough
many Desarguesian configurations, then under a technical condition the scheme
is schurian and separable. This result enables us to give short proofs for
known statements on the schurity and separability of quasi-thin and
pseudocyclic schemes. Moreover, by the same technique we prove a new result:
given a prime $p$, any $\{1,p\}$-scheme with thin residue isomorphic to an
elementary abelian $p$-group of rank greater than two, is schurian and
separable.
