Generating integrable systems from an autonomous ordinary differential equation

Abstract: The freedom to realise special-function dynamics within integrable Kummer-type Hamiltonian systems remains underexplored. We investigate this freedom through examples constructed from prescribed autonomous scalar equations, using the established reduction and mode-reconstruction framework. A local holomorphic equation $ρ^\prime = f(ρ)$ determines a family of canonical Hamiltonians on a selected energy level, subject to regularity and compatible branch choices. We introduce the category of log-augmented polynomial systems, whose polynomial mode equations retain logarithmic terms in their scalar reductions, and illustrate it with examples including the Lotka-Volterra model and a parity-time-symmetric dimer. For rational generating equations, we prove a correspondence with extended Lambert-W-type inverse laws and establish elementary evaluation of rational reconstruction quadratures, giving local solvability results for the associated coupled systems. Further constructions use inverse incomplete beta and inverse spherical Bessel functions and the modular $λ$ function. The modular example has explicit theta-constant mode solutions and a direct map to the Darboux-Halphen system.