Spaces of constant curvatere, the Liouville equation and contractions of Lie algebras

The Liouville equation is equivalent to the two-dimensional Laplace equation and is reduced to it by the Lie-Bäcklund transformation, which, from the point of view of group theory, can be interpreted from using the Inönü-Wigner contraction. We analyze the construction of solutions to the Liouville equation to represent zero curvature using the contraction of the Lie algebra sl(2) and use nilpotent generators for this.