ANZIAM  J.  44 (2002), 141-148
On the geometry of the Painlevé V equation and a Bäcklund transformation

W. K. Schief
  School of Mathematics
  The University of New South Wales
  Sydney NSW 2052
  Australia
 


Abstract
It is shown that an integrable class of helicoidal surfaces in Euclidean space $\mathbb{E}^3$ is governed by the Painlevé V equation with four arbitrary parameters. A connection with sphere congruences is exploited to construct in a purely geometric manner an associated Bäcklund transformation.
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