In the framework of geometrical optics, we consider the inverse problem consisting in obtaining refractive-index distributions $n=n(u,v)$ of a two-dimensional transparent inhomogeneous isotropic medium from a known family $f(u,v)=c$ of monochromatic light rays, lying on a given regular surface. Using some basic concepts of differential geometry, we establish a first-order linear partial differential equation relating the assigned family of light rays with all possible refractive-index profiles compatible with this family. In particular, we study the refractive-index distribution producing, as light rays, a given family of geodesic lines on some remarkable surfaces. We give appropriate examples to explain the theory.