Involutive scroll structures on solutions of 4D dispersionless integrable hierarchies

A rational normal scroll structure on an (n+1)-dimensional  manifold M is defined as a field of rational normal scrolls of degree n-1 in the projectivised cotangent bundle PT*M.
We show that geometry of this kind naturally arises on solutions  of various 4D dispersionless integrable hierarchies of  heavenly type equations. In this context, rational normal scrolls  coincide with the characteristic varieties (principal symbols) of the hierarchy. Furthermore, such structures automatically satisfy an additional property of involutivity.
Our main result states that involutive scroll structures are themselves  governed by a dispersionless integrable hierarchy, namely, the hierarchy of conformal self-duality equations.
Based on joint work with Boris Kruglikov.