Subalgebras of integrals, commutants, and superintegrable deformations of Lotka-Volterra systems

TL;DR

This paper introduces a method to deform integrable and superintegrable Lotka-Volterra systems using Poisson algebra subalgebras, preserving integrability and expanding the class of solvable models.

Contribution

It develops a Lie-algebraic framework for deforming Hamiltonian systems via subalgebras of integrals, enabling the construction of new superintegrable systems in arbitrary dimensions.

Findings

Superintegrability is preserved under the deformation process.

Explicit constructions for Abelian and non-Abelian subalgebras are provided.

Deformed systems include polynomial to rational transformations, some with non-rational integrals.

Abstract

We consider the Lie-algebraic notion of commutant in the setting of Poisson algebra. This provides a framework for deforming Hamiltonian differential equations. By taking a subalgebra of the algebra of integrals, and considering the set of functions that Poisson commute with that subalgebra, the Hamiltonian can be deformed, while retaining integrability. We deform Liouville integrable and superintegrable Lotka-Volterra systems studied in [19]. We present different explicit constructions considering Abelian and non-Abelian subalgebras of integrals. We obtain superintegrable systems for specific dimensions, and in arbitrary dimension. Polynomial systems are deformed to rational systems, some of which have non-rational integrals. Superintegrability seems to be preserved in this approach.