Conformally invariant discrete elliptic Liouville equation

The complex Liouville equation as well as the real hyperbolic one are invariant under the direct product of two Virasoro groups. The Lie algebra of the real elliptic Liouville equation does not have the structure of a direct sum. We discretize this equation on a lattice that is rotationally symmetric. The obtained difference system is invariant under the simple Lie group O(3,1). This is the maximal finite-dimensional subgroup of the infinite dimensional symmetry group of the differential equation and figures here as the conformal group of the space E2. Some thoughts on the general problem of symmetry preserving discretizations of PDES with infinite dimensional symmetry groups are presented.