Exact solutions of the d'Alembert and Liouville equations in the pseudo- Euclidean space \(R_{2,2}\). II (Q1814614)

[For part I, cf. Ukr. Math. J. 42, No. 8, 1001-1006(1990); translation from Ukr. Mat. Zh. 42, No. 8, 1122-1128 (1990; Zbl 0705.35083)] In the theory of surfaces in Euclidean space a metric of the form \(ds^ 2=g(z,\bar z) dz d\bar z\) (where \(g(z,\bar z)>0)\) is introduced, such that such surface becomes isomorphically isometric to the Lobachevskian plane. Since \(g(z,\bar z)>0\), one can introduce a function \(\varphi\) such that \(g(z,\bar z)=e^{\varphi(z,\bar z)}\). The Gaussian curvature \(K\) is given by \(K=-2e^{-\varphi}(\partial^ 2\varphi/\partial z\partial\bar z)\), or if \(K\) is constant: \(\Delta\varphi=-\Lambda e^ \varphi\) where \(\Lambda=2K\), which is Liouville's equation. Closely related is the d'Alembert equation \(\Delta\varphi=-\lambda\varphi^ K\). The authors study the invariants of maximal subalgebras of the Poincaré algebra \(A\tilde P(2,2)\). They conduct a lengthy and detailed computation reducing these equations with respect to a maximal subalgebra of rank 3, and deriving exact solutions to both equations.