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

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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