Steady-state oscillations in continuously inhomogeneous medium described by a generalized Darboux equation

We refine the result of Ovsyannikov on the general form of second order linear differential equations with a nonzero generalized Laplace invariant admitting a Lie group of transformations of maximal order with $n>2$ independent variables for which the associated Riemannian spaces have nonzero curvature. We show that the set of these equations is exhausted by the generalized Darboux equation and the Ovsyannikov equation. We find the operators acting on the set of solutions inside every one-parameter family of generalized Darboux equations. For the elliptic generalized Darboux equation possessing the maximal symmetry and describing steady-state oscillations in continuously inhomogeneous medium with a degeneration hyperplane, by group analysis methods we obtain exact solutions to boundary value problems for certain domains (generalized Poisson formulas), which in particular can be test solutions in simulating steady-state oscillations in continuously inhomogeneous media.