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Surfaces in projective differential geometry and
integrable systems

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*E. V. Ferapontov*

**E-mail:** fer@landau.ac.ru
**Abstract.** Our aim here is to give a brief review of surfaces
in projective 3-space with the emphasize on the nonlinear equations underlying
them. As examples we have choosen isothermally asymptotic surfaces, projectively
applicable surfaces, surfaces of Jonas and projectively minimal surfaces.
It is demonstrated, that the corresponding projective `Gauss-Codazzi' equations
reduce to integrable systems which are familiar from the modern soliton
theory and coincide with the stationary flows in the Davey-Stewartson and
Kadomtsev-Petviashvili hierarchies, equations of the Toda lattice, etc.
The corresponding Lax pairs can be obtained by inserting a spectral parameter
in the equations of the Wilczynski moving frame.

**AMSclassification.** 53A40, 53A20, 35Q58, 35L60

**Keywords.** Projective differential geometry, integrable systems