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Einstein-Weyl geometry, the dKP equation and twistor theory. (English) Zbl 0990.53052
The authors study the Einstein-Weyl (EW) equations in relation to integrable systems, and in particular the dispersionless Kadomtsev-Petviashvili (dKP) equation. They construct and characterise a class of new EW structures in $$2+1$$ dimensions out of solutions to the dKP equation. As a result they show that the dKP solutions give rise to hyper-Kähler metrics in four dimensions. Moreover they construct some new examples of EW structures and obtain all solutions of the dKP equation with the property that the associated EW space admits a family of divergence-free, shear-free geodesic congruences. These solutions give rise to new EW metrics depending on the arbitrary function of one variable.

##### MSC:
 53C28 Twistor methods in differential geometry 32L25 Twistor theory, double fibrations (complex-analytic aspects) 37K25 Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with topology, geometry and differential geometry 35Q55 NLS equations (nonlinear Schrödinger equations) 53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.) 83C60 Spinor and twistor methods in general relativity and gravitational theory; Newman-Penrose formalism
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