In the framework of geometrical optics we consider a two-dimensional transparent inhomogeneous isotropic medium (dispersive or not). We show that (i) for any family belonging to a certain class of planar monoparametric families of monochromatic light rays given in the form $f(x,y)=c$ of any definite color and satisfying a differential condition, all the refractive index profiles $n=n(x,y)$ allowing for the creation of the given family can be found analytically (inverse problem) and that (ii) for any member of a class of two-dimensional refractive index profiles $n=n(x,y)$ satisfying a differential condition, all the compatible families of light rays can be found analytically (direct problem). We present appropriate examples.