The two-dimensional eikonal equation

We study the two-dimensional eikonal equation ψ _x ² + ψ _y ² = 1/v ²(x, y). We carry out the group analysis of the equation, establish a connection between the group properties and geometric characteristics of the Riemannian space with the metric ds ² = [dx ² + dy ²]/v ²(x, y). We select the most important classes of equations and derive some conditions for reducibility of a given equation to an equation of one of those classes. We find a condition for two equations to be equivalent (the theorem of seven invariants). For the equations corresponding to Riemannian spaces of constant curvature, we obtain explicit formulas for the solutions describing the wave front for a point source and also the ray equations.