Design considerations for two-dimensional symmetric bootlace lenses

It is generally known that a symmetrical bootlace lens having two pairs of conjugate foci can be designed. Additionally, there is a bootlace lens, the R-2R, which focuses perfectly at infinitely many points. It is shown in this paper that the R-2R is unique; there are no other perfect bootlace lenses and, in fact, no others with more than two pairs of conjugate off-axis foci.