Bi-presymplectic representation of Liouville integrable systems and related separability theory

Bi-presymplectic chains of one-forms of arbitrary co-rank are considered. The conditions in which such chains represent some Liouville integrable systems and the conditions in which there exist related bi-Hamiltonian chains of vector fields are presented. In order to derived the construction of bi-presymplectic chains, the notions of dual Poisson-presymplectic pair, d-compatibility of presymplectic forms and d-compatibility of Poisson bivectors is used. The completely algorithmic construction of separation coordinates is demonstrated. It is also proved that St"{a}ckel separable systems have bi-inverse-Hamiltonian representation, i.e. are represented by bi-presymplectic chains of closed one-forms. The co-rank of related structures depends on the explicit form of separation relations.

💡 Research Summary

The paper introduces a novel geometric framework for Liouville‑integrable systems based on “bi‑presymplectic chains” of one‑forms whose co‑rank may be arbitrary. Starting from the classical bi‑Hamiltonian theory, the authors generalize the notion of a symplectic‑Poisson pair to a dual Poisson‑presymplectic pair (Π, ω) where Π is a Poisson bivector and ω a possibly degenerate (presymplectic) two‑form. The duality condition is expressed as Π·ω = Id − Pₖ, with Pₖ the projector onto the common kernel of dimension k (the co‑rank). This relation reduces to the usual inverse when k = 0 but remains meaningful for degenerate structures.

A central concept is d‑compatibility. Two presymplectic forms ω₁, ω₂ are d‑compatible if both are closed (dωᵢ = 0) and their difference is exact, which guarantees that they share the same cohomology class. Analogously, two Poisson bivectors Π₁, Π₂ are d‑compatible when their Schouten–Nijenhuis bracket is d‑exact. Under d‑compatibility, one can construct a bi‑Hamiltonian chain \