Necessary condition for symmetry-integrability of odd-order evolution equations (Conjecture 1)

Prove that for every integer k ≥ 1, any odd-order evolution equation of the form u_t = F(x, t, u, u_x, u_{xx}, …, u_{(2k+3)x}) that is symmetry-integrable (i.e., admits infinitely many local Lie–Bäcklund symmetry generators) must satisfy the differential relation (2k+3)·(∂F/∂u_{(2k+3)x})·(∂^3F/∂u_{(2k+3)x}^3) − (3k+5)·(∂^2F/∂u_{(2k+3)x}^2)^2 = 0.

Background

Section 3 introduces a conjectured necessary condition for symmetry-integrability of odd-order (2k+3) evolution equations. The authors derive this from extensive computations of Lie–Bäcklund symmetries up to order 19 and focus on equations relevant to the two sequences defined earlier in Proposition 1.

They emphasize that the 3rd-order case (u_t = F(x, t, u, u_x, u_{xx}, u_{3x})) is special and governed by a different, known 4th-order condition, whereas Conjecture 1 addresses odd orders ≥ 5. The conjecture provides an explicit relation involving partial derivatives of F with respect to the highest derivative u_{(2k+3)x}, intended as a necessary condition for the existence of infinitely many local Lie–Bäcklund symmetries.