Design of a stigmatic lens implementing a required ray mapping

We consider a method for designing stigmatic lenses implementing required ray mappings. A ray mapping relates the angular coordinates of the rays at the image points with the angular coordinates of the rays coming from the object. The calculation of a stigmatic lens with a required ray mapping is reduced to solving an explicit ordinary differential equation. As examples, we design stigmatic lenses satisfying the Abbe sine condition, the Herschel's condition, and the constant angular magnification condition.