A GL(2,[special characters omitted])-structure on a smooth manifold of dimension n+1 corresponds to a distribution of non-degenerate rational normal cones over the manifold. Such a structure is called k-integrable if there exist many foliations by submanifolds of dimension k whose tangent spaces are spanned by vectors in the cones.

This structure was first studied by Bryant (n = 3 and k = 2). The work included here (n = 4 and k = 2, 3) was suggested by Ferapontov, et al., who showed that the cases (n = 4, k = 2) and (n = 4, k = 3) can arise from integrability of second-order PDEs via hydrodynamic reductions.

Cartan-Kähler analysis for (n = 4, k = 3) leads to a complete classification of local structures into 55 equivalence classes determined by the value of an essential 9-dimensional representation of torsion for the GL(2,[special characters omitted])-structure. These classes are described by the factorization root-types of real binary octic polynomials. Each of these classes must arise from a PDE, but many of the PDEs remain to be identified.

Also, we study the local problem for n ≥ 5 and k = 2, 3 and conjecture that similar classifications also exist for these cases; however, the most interesting results are essentially unique to degree 4. The approach is that of moving frames, using Cartan's method of equivalence, the Cartan-Kähler theorem, and Cartan's structure theorem.