A new family of integrable differential systems in arbitrary dimension
J. D. García-Saldaña, A. Gasull, S. Rebollo-Perdomo
TL;DR
The paper develops a universal method for generating completely integrable differential systems in arbitrary dimension by using a dimension‑preserving, potentially non‑invertible map $\bm{\Phi}$ to pull back integrals from a reduced system $du/dt = G(u)$. The core result shows that if the original vector field can be written in terms of $\bm{\Phi}$ as $F(x) = R(x)\,\mathrm{adj}(D\Phi(x))\,G(\Phi(x))$, then every first integral $I$ of the reduced system yields a first integral $H(x)=I(\Phi(x))$ of the original, and complete integrability follows when $\mathbf{G}(0)\neq 0$ or the reduced system is completely integrable. The authors provide extensive plane, space, and higher‑dimensional examples (including Kolmogorov, Lotka–Volterra, Liénard, Loud, Rössler, Rikitake, and nilpotent families) to illustrate how polynomial or analytic first integrals arise as pulled‑back functions. This framework extends known planar integrability results to higher dimensions and offers a versatile mechanism to generate new integrable systems with singularities, with potential implications for qualitative analysis and explicit integration.
Abstract
We present a wide class of differential systems in any dimension that are either integrable or complete integrable. In particular, our result enlarges a known family of planar integrable systems. We give an extensive list of examples that illustrates the applicability of our approach. For instance, in the plane this list includes some Liénard, Lotka--Volterra and quadratic systems; in the space, some Kolmogorov, Rikitake and Rössler systems. Examples of complete integrable systems in higher dimensions are also provided.