For several classes of second order dispersionless PDEs, we show that the
symbols of their formal linearizations define conformal structures which must
be Einstein-Weyl in 3D (or self-dual in 4D) if and only if the PDE is
integrable by the method of hydrodynamic reductions. This demonstrates that the
integrability of these dispersionless PDEs can be seen from the geometry of
their formal linearizations.Comment: In this version we add Preliminary Section and the Appendix, where we
discuss the geometry of PDEs and the method of hydrodynamic reductions. Also
we add Lax pairs for the 5 integrable equations of type I, and supply the
ancillary files (Maple verifications of the calculations, Maple and
Mathematica form of Integrability Conditions, together with their PDF
versions) for completenes
