Classification conjecture for integrable bi-Hamiltonian equations

An integrable bi-Hamiltonian equation in one dependent variable $u$ is considered up to contact transformations. The relevant model hierarchies are the linear, KdV, and HD hierarchies, the hierarchy from Theorem 3.7(b), and those discussed in Remarks 3.10, 3.11, and 3.14. Bi-Hamiltonian hierarchy classification conjecture. Every integrable bi-Hamiltonian equation in $u$ is contact-equivalent to one contained in one of those hierarchies. This is consistent with known classification results for general integrable equations, but the asserted exhaustive classification remains open.