Lax pair and invariant manifold structures in Z-Hamiltonians for s>1

Determine whether the Z-Hamiltonian systems defined by the resonant equation i dα_n/dt = Σ_{n+m=k+j} C^{(s)}_{nmkj} α_k α_j overline{α_m} with couplings C^{(s)}_{nmkj} given in Equation (C_nmij_equation) possess Lax pair structures and multi-dimensional invariant manifolds analogous to those known for the cubic Szegő equation (s=1).

Background

The cubic Szegő equation (s=1) is integrable and admits both a Lax pair structure and an infinite hierarchy of invariant manifolds. The Z-Hamiltonian family generalizes this equation to s>1 while retaining a specific three-dimensional invariant manifold used throughout the paper to construct explicit cascade and condensation solutions.

While the s=1 case is well understood, the structural integrability properties (Lax pair and richer invariant manifolds) of the Z-Hamiltonians for s>1 are not established. Clarifying whether these systems inherit similar integrable structures is central to understanding their dynamics and potential for exact solvability beyond the one invariant manifold analyzed.