  The symmetry algebra of the real elliptic Liouville equation is an
infinite-dimensional loop algebra with the simple Lie algebra $o(3,1)$ as its
maximal finite-dimensional subalgebra. The entire algebra generates the
conformal group of the Euclidean plane $E_2$. This infinite-dimensional algebra
distinguishes the elliptic Liouville equation from the hyperbolic one with its
symmetry algebra that is the direct sum of two Virasoro algebras. Following a
discretisation procedure developed earlier, we present a difference scheme that
is invariant under the group $O(3,1)$ and has the elliptic Liouville equation
in polar coordinates as its continuous limit. The lattice is a solution of an
equation 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.
