Axisymmetric aplanatic systems have been used in the past for solar concentrators and condensers (Gordon et. al, 2010).
It is well know that such a system must be stigmatic and satisfy the Abbe sine condition. This problem is well known
(Schwarzschild, 1905) to be solvable with two aspherics when the system has rotational symmetry.
However, some of those axisymmetric solutions have intrinsically shading losses when using mirrors, which can be
prevented if freeform optical surfaces are used (Benitez, 2007).
In this paper, we explore the design of freeform surfaces to obtain full aplanatic systems. Here we prove that a rigorous
solution to the general non-symmetric problem needs at least three free form surfaces, which are solutions of a system of
partial differential equations (PDE). We also present the PDEs for a three surface full aplanat. The examples considered
have one plane of symmetry, where a consistent 2D solution is used as boundary condition for the 3D problem. We have
used the x-y polynomial representations for all the surfaces, and the iterative algorithm formulated for solving the above
said PDE has shown very fast convergence.