A Reciprocal Transformation for the Constant Astigmatism Equation
  We introduce a nonlocal transformation to generate exact solutions of the constant astigmatism equation $z_{yy} + (1/z)_{xx} + 2 = 0$. The transformation is related to the special case of the famous B\"acklund transformation of the sine-Gordon equation with the B\"acklund parameter $\lambda = \pm1$. It is also a nonlocal symmetry. 
Adam Hlav\'a\v{c} and Michal Marvan
nlin.SI math.DG