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arXiv · 2602.09747

Darboux first integrals of Kolmogorov systems with invariant $n$-sphere

Supriyo Jana·Soumen Sarkar
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Abstract

In this paper, we characterize all polynomial Kolmogorov vector fields for which the standard $n$-sphere is invariant. We exhibit completely integrable Kolmogorov vector fields of degree $m$ on $\mathbb{S}^n$ for any $m >2$. Then, we show that there is no cubic Hamiltonian Kolmogorov vector field that makes an odd-dimensional sphere invariant. We examine the conditions under which a cubic Kolmogorov vector field has a Darboux first integral. In many cases, we determine whether they constitute necessary and sufficient conditions. Moreover, we study the complete integrability of cubic Kolmogorov vector fields having an invariant $n$-sphere.

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Supriyo Jana, Soumen Sarkar. 2026-02-10. Darboux first integrals of Kolmogorov systems with invariant $n$-sphere. https://doi.org/10.1016/j.nonrwa.2026.104671

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