Lax pairs and higher-dimensional invariant manifolds in Z- and Y-Hamiltonian families

Ascertain whether the Z-Hamiltonian and Y-Hamiltonian systems possess Lax pair structures and establish whether they admit multi-dimensional invariant manifolds beyond the three-dimensional invariant manifolds analyzed in this work; alternatively, determine if the Lax pair property is exclusive to the cubic Szegő equation and the truncated Szegő equation within their respective families.

Background

The paper introduces two infinite families of resonant Hamiltonian systems (Z and Y) exhibiting explicit cascade phenomena. Within each, a specific invariant manifold enables exact solutions. The cubic Szegő equation and truncated Szegő equation—special members of these families—are known to have Lax pairs and rich invariant manifold structures.

Whether these broader families also possess Lax pair structures and additional higher-dimensional invariant manifolds remains unaddressed. Resolving this would significantly impact the integrability and analytical tractability of these systems.