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Algebraic First Integrals of the Polynomial Systems Satisfying the Cauchy–Riemann Conditions

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Abstract

We consider polynomial planar systems of ordinary differential equations that can be written in the form \(\dot{z}={\mathcal {P}}(z), \ z\in {\mathbb {C}}.\) We describe the classes of such systems that have rational first integrals and study some properties of these integrals.

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Notes

  1. If a function h(z) is written in the form \(h(z)=g(z,\bar{z})\), then the operations \(\partial /\partial z, \ \partial /\partial \bar{z}\) are reduced to finding usual partial derivatives of the function \(g(z, \bar{z})\) with respect to \(z, \bar{z}\).

  2. In [8] such points are called dipoles.

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Acknowledgments

The work was carried out with partial financial support of the Russian Foundation for Basic Research (grant 15-01-00745). I would like to thank Dr. L. Vertgeim for his help with English. I thank the reviewer for his comments on my manuscript.

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Authors and Affiliations

  1. Sobolev Institute of Mathematics, Novosibirsk, 630090, Russia

    Evgenii P. Volokitin

  2. Novosibirsk State University, Novosibirsk, 630090, Russia

    Evgenii P. Volokitin

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  1. Evgenii P. Volokitin
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Correspondence to Evgenii P. Volokitin.

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Volokitin, E.P. Algebraic First Integrals of the Polynomial Systems Satisfying the Cauchy–Riemann Conditions. Qual. Theory Dyn. Syst. 15, 575–596 (2016). https://doi.org/10.1007/s12346-015-0174-8

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