Superintegrability of a resonant ghostly tri-Hamiltonian model: polynomial first integrals and transcendental completion

Abstract: We study classical and quantum superintegrability of a resonant three-dimensional Hamiltonian with indefinite kinetic energy and three Poisson descriptions of the same flow. Besides three commuting quadratic Hamiltonians, the polynomial first-integral ring contains two primitive degree-four generators, $I_4^{\rm o}$ and $I_4^{\rm e}$, of odd and even momentum parity. A single octic relation leaves the polynomial invariant ring with transcendence degree four. Solving the complete first-integral equation shows that every rational first integral is a rational function of $H_1,H_2,H_3,I_4^{\rm o}$, so that the rational invariant field likewise has generic functional rank four. A branch-free transcendental invariant raises the generic smooth functional rank to five on the invariant open set $Δ>0$. The system is therefore minimally superintegrable in the polynomial and rational classes and maximally superintegrable on this invariant domain when transcendental integrals are admitted. Weyl quantisation yields exact higher-order differential symmetries in all three Poisson realisations. The branch-free fifth classical invariant also admits a densely defined nonlocal quantum counterpart, providing a concrete quantum realisation of the transcendental completion.