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The symmetry algebra of the real elliptic Liouville equation is aninfinite-dimensional loop algebra with the simple Lie algebra $o(3,1)$ as itsmaximal finite-dimensional subalgebra. The entire algebra generates theconformal group of the Euclidean plane $E_2$. This infinite-dimensional algebradistinguishes the elliptic Liouville equation from the hyperbolic one with itssymmetry algebra that is the direct sum of two Virasoro algebras. Following adiscretisation procedure developed earlier, we present a difference scheme thatis invariant under the group $O(3,1)$ and has the elliptic Liouville equationin polar coordinates as its continuous limit. The lattice is a solution of anequation invariant under $O(3,1)$ and is itself invariant under a subgroup of$O(3,1)$, namely the $O(2)$ rotations of the Euclidean plane.

Conformally invariant elliptic Liouville equation and its symmetry preserving discretization