First Integrals of a Cubic System of Differential Equations

Zhibek Kadyrsizova, Valery G. Romanovski, Marko Robnik

2008

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DOI 10.1063/1.3046241
Data confidence 70 medium
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Abstract

We study polynomial systems of differential equations of the form ẋ = x+p(x,y), ẏ = −3y+q(x,y), (1) where p(x,y) and q(x,y) are homogeneous polynomials of degree three. In [1] the linearizability problem for this system has been studied. In many cases considered in [1] the construction of linearizing transformations requires knowledge of first integrals. In [2] it has been proved that such integrals exist. In the present paper we find the explicit expressions for the integrals. It allows to obtain the explicit formulas for the corresponding linearizing transformations of [1].