Recursion operator for the fully nonlinear third-order Schwarzian equation in Case 1.4.3

Construct a recursion operator for the fully nonlinear third-order Schwarzian equation with characteristic function \(\Phi(S;a_1,a_2)=(a_1-S)/( (a_1^2+3a_2)(S^2-2a_1S-3a_2)^{1/2})\), subject to \(a_1^2+3a_2\neq 0\), in order to generate its hierarchy of fully nonlinear Schwarzian equations directly in the dependent variable \(u\).

Background

Case 1.4.3 considers a fully nonlinear third-order Schwarzian evolution equation whose characteristic function depends on the parameters $a_1$ and $a_2$. The authors report that a sixth-order recursion operator is known for the corresponding auxiliary $S$-equation, which can be used to generate the hierarchy in the $u$-variable indirectly.

The unresolved issue is the construction of a recursion operator for the fully nonlinear equation itself. Such an operator would provide a direct mechanism for generating the equation's higher-order symmetry hierarchy.