Uniqueness/nonexistence of equilibrium solutions in canonical coordinates for bounded vs. unbounded motion

Establish whether, for the N-species integrable Volterra system with interaction matrix A_{rs} = ε_r ε_s (B_r − B_s) and canonical coordinates (P,Q), the equilibrium equations 1 − Σ_{k=1}^N η_k exp(P^0 ε_k + Q^0 η_k) 𝒞_k = 0 and Σ_{k=1}^N ε_k exp(P^0 ε_k + Q^0 η_k) 𝒞_k = 0 (where 𝒞_k = exp(Σ_{j=1}^{N−2} R_j τ^{(j)}_k) are exponentials of Casimir combinations) admit exactly one solution (P^0,Q^0) when trajectories are bounded and admit no solutions when trajectories are unbounded, as a function of the parameter values (ε_k, B_k) and the Casimir constants R_j. Provide a proof or a counterexample of this conjectured characterization.

Background

The paper reduces Volterra’s N-species integrable Lotka–Volterra model with rank-2 interaction matrix A_{rs} = ε_r ε_s (B_r − B_s) to a Hamiltonian system with one degree of freedom, using canonical variables (P,Q) (and alternatively (p,q)). Equilibrium configurations in canonical coordinates are defined by the pair of equations 0 = 1 − Σ η_k exp(P^0 ε_k + Q^0 η_k) 𝒞_k and 0 = Σ ε_k exp(P^0 ε_k + Q^0 η_k) 𝒞_k, with 𝒞_k fixed by Casimirs of the underlying Poisson structure.

Numerical evidence in the paper suggests that, in the (P,Q) plane, bounded (periodic) motions appear to have a single equilibrium configuration, whereas unbounded motions appear to have none. The authors therefore conjecture a precise dichotomy—exactly one equilibrium solution for bounded motion and zero for unbounded motion—but explicitly state they cannot provide a proof, inviting a rigorous resolution of this question.