Existence of bi-chromatic 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 surface rays will be dispersed inside the material. We will study the existence of a lens composed of two surfaces such that this dispersion is eliminated when leaving toward the 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 elab-orated and requires the study of a functional system of differential equations of the form

We will apply an approach similar to the one of Picard and Lindelof for ODEs and show that under some Lipschitz condition on H a solution to the system exists.