Conjecture: Classification of all symmetry‑integrable projective‑invariant evolution equations

Prove or disprove that every symmetry‑integrable scalar evolution equation in one spatial dimension that is invariant under the projective (Möbius) transformation acting as u → (a1 u + b1)/(c1 u + d1), x → (a2 x + b2)/(c2 x + d2), and t → t + ε with a1 d1 − b1 c1 = 1 and a2 d2 − b2 c2 = 1 belongs to the hierarchy generated by the fully nonlinear third‑order Möbius‑invariant equation u_t = λ u_x S[u]^{-1/2}, where S[u] = u_{xxx}/u_x − (3/2)(u_{xx}/u_x)^2 denotes the Schwarzian derivative.

Background

The paper constructs invariants for a projective transformation that acts Möbius‑linearly and independently on both the dependent variable u(x,t) and the spatial variable x, together with a translation in t. Using these invariants, the authors enumerate all invariant evolution equations up to order seven.

They identify a fully nonlinear third‑order equation that is symmetry‑integrable, exhibit its recursion operator, and derive its hierarchy. For the fifth‑order case, they prove that the only symmetry‑integrable equation in the invariant class lies in this hierarchy. For sixth‑ and seventh‑order cases, they did not find exceptions but could not prove full generality, leading them to conjecture that every symmetry‑integrable projective‑invariant evolution equation falls within the hierarchy generated by the third‑order equation.