  We construct a family of second-order linear difference equations
parametrized by the hypergeometric solution of the elliptic Painlev\'e equation
(or higher-order analogues), and admitting a large family of
monodromy-preserving deformations. The solutions are certain semiclassical
biorthogonal functions (and their Cauchy transforms), biorthogonal with respect
to higher-order analogues of Spiridonov's elliptic beta integral.
