  Exploiting the description of rings of differential operators as Azumaya
algebras on cotangent bundles, we show that the moduli stack of flat
connections on a curve (allowed to acquire orbifold points) defined over an
algebraically closed field of positive characteristic is \'etale locally
equivalent to a moduli stack of Higgs bundles over the Hitchin base. We then
study the interplay with stability and generalize a result of Laszlo-Pauly,
concerning properness of the Hitchin map. Using Arinkin's autoduality of
compactified Jacobians we extend the main result of Bezrukavnikov-Braverman,
the Langlands correspondence for D-modules in positive characteristic for
smooth spectral curves, to the locus of integral spectral curves.
