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

2026/02/20 by Ian Marquette, Peter H. van der Kamp, G. R. W. Quispel · 1 voice
Physics and Astronomy · #nlin.SI

paper · pdf

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.

Citations

• On the construction of polynomial Poisson algebras: a novel grading approach
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• A-Poisson structures
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Discussions

• Ian Marquette, Peter H. van der Kamp, G. R. W. Quispel: Subalgebras of integrals, commutants, and superintegrable deformations of Lotka-Volterra systems https://arxiv.org/abs/2602.17945 https://arxiv. [bsky, 0 points, 0 comments]

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