Partial integrals and the first focal value in the problem of centre
The authors prove that for planar polynomial differential systems of degree \(n\geq 3\) if \(O(0,0)\) is weak focus of order \(\geq 2\), and there exist \(N= (n^ 2+ n- 4)/2\) partial algebraical integrals, then \(O\) is a centre. For cubic systems they obtain 14 series of conditions each ensuring that the origin is a centre. In all these cases the partial integrals are all straight lines. They also give examples to show that for cubic systems sometimes the number of integral straight lines can be \(<4\) in order to ensure \(O(0,0)\) to be a centre; but sometimes on increasing the number of focal values equal to zero along with a corresponding decrease of the number of integral straight lines does not always imply the existence of a centre at \(O\). Notice that for \(n= 2\) \((N= 1)\), assuming that the partial integral is a straight line, a Chinese mathematician N. D. Zhu has already proved in 1965 that \(O\) is a centre. On discussing a similar problem, we refer the reader to a recent paper of \textit{R. E. Kooij} [J. Differ. Equations 116, No. 2, 249-264 (1995)].
scientific article; zbMATH DE number 736848