Characterization of rational and polynomial Kummer-type Hamiltonians

Characterize all Kummer-type Hamiltonians whose coupled mode equations are rational or polynomial, while specifying the canonical variables and the allowable reparameterizations of the independent variable.

Background

The paper constructs integrable Kummer-type Hamiltonian systems from prescribed autonomous scalar equations and exhibits mechanisms producing rational or polynomial mode equations, including nonrational Hamiltonians and local time reparameterizations. It emphasizes that polynomiality of the coupled equations does not imply that the associated scalar reduction is elliptic or hyperelliptic, particularly for the log-augmented examples based on Lotka–Volterra dynamics and related systems.

The authors identify a broader unresolved classification issue: determining the complete class of Hamiltonians yielding rational or polynomial mode equations requires accounting simultaneously for the choice of canonical variables and for allowable changes of the independent variable. The paper provides sufficient constructions but does not establish a complete characterization.