In two dimensions, the group of conformal automorphisms of a space can be quite large (as in the case of Lorentzian signature) or variable (as with the case of Euclidean signature). The comparative lack of rigidity of the two-dimensional case with that of higher dimensions owes to the analytical fact that the asymptotic developments of the infinitesimal automorphisms of the structure are relatively unconstrained. In Lorentzian signature, the freedom is in a pair of real valued functions. In Euclidean, the freedom is in a single holomorphic function.
The stabilizer of the null ray pointing up the last coordinate vector is given by the Borel subalgebra