  We study the singularities of algebraic difference equations on curves from
the point of view of equivariant sheaves. We propose a definition for the
formal local type of an equivariant sheaf at a point in the case of a reduced
curve acted on by a group which is virtually the integers. We show that with
this definition, equivariant sheaves can be glued from an "open cover".
Precisely, we show that an equivariant sheaf can be uniquely recovered from the
following data: the restriction to the complement of a point, the local type at
the point itself, and an isomorphism between the two over the punctured
neighborhood of said point.
  We study symmetric elliptic difference equations ("elliptic equations" from
now on) from this point of view. We consider several natural notions for an
algebraic version of symmetric elliptic difference equations, i.e. symmetric
elliptic difference modules ("elliptic modules"). We show that different
versions are not equivalent, but we detail how they are related: all the
versions embed fully faithfully into the same category of equivariant sheaves.
This implies that we can use the theory for equivariant sheaves to study
singularities of elliptic equations as well.
  One reason to study elliptic equations is that they generalize, and
degenerate to, (q-)difference equations (i.e. equivariant sheaves) and
differential equations (i.e. D-modules) on the projective line. We discuss this
from the elliptic module point of view, which requires studying elliptic
modules on singular curves. We study the relation between elliptic modules on
singular curves and their normalization. We show that for modules which are
flat at the singular points there is an equivalence and we give examples
showing that this cannot be improved upon.
