Two-dimensional classical superintegrable systems: polynomial algebra of integrals

Abstract We present a constructive framework for deriving Poisson-algebraic structures in classical two-dimensional superintegrable Hamiltonian systems, characterized by three functionally independent integrals of motion H , L , and A . Under explicit geometric hypotheses—namely, that one can select an integral L whose Hamiltonian flow is complete and 2 π -periodic on each connected regular region—we obtain globally defined generators H , L , A , and B = { L , A } closing to form a four-generated Poisson algebra. The construction is fully explicit and does not require the integrals to be polynomial in the momenta, of bounded differential order, or associated with separable coordinates. Polynomial closure follows whenever the corresponding structure function G ( H , L ) is polynomial in H and L . This provides a general mechanism for constructing polynomial Poisson algebras well beyond the standard polynomial-momentum setting, including systems with higher-order and non-polynomial integrals of motion. We illustrate the method in several relevant examples, including the Kepler, Holt, Smorodinsky–Winternitz, Fokas–Lagerstrom, Hamiltonian trigonometric in the momenta, non-separable Post–Winternitz, and curved-space oscillator systems. In several cases, the same set of integrals also allows one to reconstruct the trajectories in configuration space by algebraic elimination of the momenta, without direct integration of Hamilton’s equations. We also discuss distinguished invariant sectors associated with special values of the integrals of motion.