On the existence of bichromatic lenses

We consider the following optical setting. Rays are emitted from infinity (far field) or from a point source(near field). The rays are assumed bichromatic i.e composed of two colors. Having a refractive lens rays will be dispersed inside the material. We will study the existence of a lens such that this dispersion is eliminated when leaving toward he target, i.e. all the colors shall leave the upper face with the same direction.

To study the far field case, we use a fixed point type argument to show (non)existence of such surfaces. The near field problem however is more elaborated and requires the study of a functional system of ODE of the form
Z' (t) = H(t; Z(t), Z(z_1 (t)); Z' (t), Z ' (z_1 (t))),        Z(0) = 0

We will use the approach of Picard and Lindelof for ODEs and show thatunder some Lipschitizity condition on H a solution to the system exist.