  We classify the reflexive modules of rank one over rational and minimally
elliptic singularities. Equivalently, we classify full line bundles on the
resolutions of rational and minimally elliptic singularities. As an
application, we determine among such reflexive modules of rank one all the
special ones (in the sense of Wunram) and all the flat ones. In this way, we
also classify the non-flat reflexive modules as well (as a generalization of a
construction of Dan and Romano).
  In particular, we prove (in the rank one case) a conjecture of Behnke, namely
that in the case of a cusp singularity any reflexive module admits a flat
connection.
  The results generalize the classical Mckay correspondence, and results of
Artin, Verdier, Esnault, Khan and Wunram valid for different particular
families of singularities.
