bibkey: marquette2026lotkavolterra
authors: Ian Marquette; Peter H. van der Kamp; G. R. W. Quispel
year: 2026
title: "Subalgebras of integrals, commutants, and superintegrable deformations of Lotka-Volterra systems"
doi: null
url: https://arxiv.org/abs/2602.17945v1
claim: "For the n-dimensional Lotka-Volterra system with integrals F_k and their coordinate reversals G_k, the change of variables from x_1,...,x_n to the n-1 distinct integrals and the distinguished variable x_n is conjectured to be birational in every dimension."
strata_touched:
  - D5/S3/FluidDynamics/Solitons/LotkaVolterraIntegralBirationality
license: citation-only
triage: anchor

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

Ian Marquette, Peter H. van der Kamp, G. R. W. Quispel, arXiv:2602.17945v1 (nlin.SI), 2026-02-20. Quotations are from the v1 TeX source.

The conjecture (Section 2):

In arbitrary dimensions, we have not been able to prove the birationality of the change of variables, but we conjecture this to be the case, and we provide explicit formulas in Appendix \ref{secA}.

The integrals (Section 3), for $n=2m-1$ and $n=2m$ respectively:

F_k:= \sum_{\ell=1}^{2k-1} x_\ell \prod_{s=k}^{(n-1)/2} \frac{x_{2s+1}}{x_{2s}},\quad k=1,\ldots,m F_k:= \sum_{\ell=1}^{2k} x_\ell \prod_{s=k}^{n/2-1} \frac{x_{2s+2}}{x_{2s+1}},\quad k=1,\ldots,m This other set is given by $G_k:=ı(F_k)$, where $k=1,\dots ,m$. We have $F_m=G_m=H$, and, when $n$ is odd, $F_1=G_1=C$

The conjecture restated (Appendix A):

We conjecture that the transformation from the $n$ variables $x_i$ to the $n-1$ integrals and distinguished variable $x_n$ is a birational map. We treat the odd and the even dimensional case separately.

Verified locator

• URL: https://arxiv.org/abs/2602.17945v1 (v1, 2026-02-20, the only version; source Deformation.tex, md5 31a9b6c519bf401c2db3f095fa8ee363): the Section 2 conjecture (l. 271), the integrals (l. 300–316) and Appendix A (from l. 1145).