  By analogy with work of Hitchin on integrable systems, we construct natural
relaxations of several kinds of moduli spaces of difference equations, with
special attention to a particular class of difference equations on an elliptic
curve (arising in the theory of elliptic special functions). The common feature
of the relaxations is that they can be identified with moduli spaces of sheaves
on rational surfaces. Not only does this make various natural questions become
purely geometric (rigid equations correspond to -2-curves), it also establishes
a number of nontrivial correspondences between different moduli spaces, since a
given moduli space of sheaves is typically the relaxation of infinitely many
moduli spaces of equations. In the process of understanding this, we also
consider a number of purely geometric questions about rational surfaces with
anticanonical curves; e.g., we give an essentially combinatorial algorithm for
testing whether a given divisor is the class of a -2-curve or is effective with
generically integral representative.
