Existence of seventh-order Schwarzian equations outside known tau-equation hierarchies

Determine whether there exist seventh-order Schwarzian evolution equations that are not \(\tau\)-equations associated with third-order or fifth-order Schwarzian equations.

Background

The paper constructs and classifies fifth-order Schwarzian equations and their seventh-order Lie–Bäcklund symmetries. The authors observe that, at present, every known seventh-order Schwarzian evolution equation is a $\tau$-equation belonging to a hierarchy generated from a third-order or fifth-order Schwarzian equation.

Whether genuinely new seventh-order equations exist outside those established hierarchies remains unresolved. Resolving this question would extend the classification of symmetry-integrable, Möbius-invariant Schwarzian evolution equations.