bᵏ-algebroids and the variety of foliation jets

We introduce and classify singular foliations of bᵏ⁺¹-type, which formalize the properties of vector fields that are tangent to a submanifold W ⊂ M to order k. When W is a hypersurface, these structures are Lie algebroids generalizing the bᵏ⁺¹-tangent bundles introduced by Scott. We prove that singular foliations of bᵏ⁺¹-type are encoded by k-th order foliations: jets of distributions that are involutive up to order k, equivalently described as foliations on the k-th order neighborhood of W. Using this encoding, we construct topological groupoids of k-th order foliations and employ the holonomy invariant to show that these groupoids fiber over certain character stacks, yielding Riemann-Hilbert style classifications up to local isomorphism and isotopy. We also study the problem of extending a k-th order foliation to a $(k+1)$-st order foliation. We prove that this is obstructed by a characteristic class that arises as a section of a vector bundle over the relevant character stack.