  We extend the recent paradigm ''Integrability via Geometry'' from dimensions
3 and 4 to higher dimensions, relating dispersionless integrability of partial
differential equations to curvature constraints of the background geometry. We
observe that in higher dimensions on any solution manifold the symbol defines a
vector distribution equipped with a subconformal structure, and the
integrability imposes a certain compatibility between them.
  In dimension 5 we express dispersionless integrability via the vanishing of a
certain curvature of this subconformal structure. We also obtain a ''master
equation'' governing all second order dispersionless integrable equations in
5D, and count their functional dimension.
  It turns out that the obtained background geometry is parabolic of the type
$(A_3,P_{13})$ and we provide its Cartan theoretic description and compute the
harmonic curvature components via the Kostant theorem. Then we relate it to 3D
projective and 4D conformal geometries via twistor theory, discuss nested
symmetry reductions, as well as give another interpretation of dispersionless
integrability in 5D through Levi-degenerate CR structures in dimension 7.
