Nonlinear realization of the conformal group in two dimensions and the Liouville equation

It is shown that the Liouville equation $u_{+-}=m^2e^{-2u}$ has an adequate description in the language of the nonlinear realization of the infinite-parameter conformal group $G$ in two dimensions. The coordinates $x^+$, $x^-$ of the two-dimensional Minkowski space and the field $u(x)$ are identified with certain parameters of the factor space $G/H$, where $H=SO(1,1)$ is the Lorentz group in two dimensions. The Liouville equation arises as one of the covariant conditions of reduction of the factor space $G/H$ to its connected geodesic subspace $SL(2,R)/H$. The alternative reduction to the subspace $\mathscr P(1,1)/H$ where $\mathscr P(1,1)$ is the two-dimensional Poincaré group, leads to the free equation for $u(x)$. The corresponding representations of zero curvature and B cklund transformations acquire in the present approach a simple group-theoretical meaning. The possibility of generalizing the proposed construction to other integrable systems is discussed.