Dispersionless integrable systems in dimension 5

Dispersionless integrability has been related to integrable background geometry via canonical structure on solutions for systems with quadratic characteristic variety (for instance, second order scalar PDEs). It turns out that the existence of dispersionless Lax pair implies the restriction that this variety is necessary degenerate if the number of independent variables exceeds four, consequently no convenient conformal metric can exist. It will be explained that the proper higher-dimensional analog is a compatible subconformal structure with the zero-curvature restriction. In this talk we will focus on dimension five, where the corresponding background geometry is either Levi-indefinite almost CR or almost para-CR, and the corresponding curvatures are given by the general theory of regular normal parabolic structures. The work is joint with Omid Makhmali.