Group classification of the eikonal equation for a 3-dimensional inhomogeneous medium

The equation (delpsi)(2) = 1/v(2)(x, y, z), known as the eikonal equation, is studied. This is the characteristic equation for the wave equations in an inhomogeneous medium, which plays a central role in the description of the geometry of the rays and the wave fronts. A full geometric classification of the family of eikonal equations is carried out (an equation is determined by the function v(x, y, z), which has the meaning of the propagation velocity of a perturbation in the medium). In the cases of equations with linear or quadratic velocity function v(x, y, z), explicit solutions - point source eikonals - are presented and the geometry of the rays is completely described.