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$1:-3$ resonant centers on ${ℂ}^{2}$ with homogeneous cubic nonlinearities. (English) Zbl 1165.34334
Summary: We characterize the collection of systems of differential equations on ${ℂ}^{2}$ of the form ${x}^{\text{'}}=x+p\left(x,y\right)$, ${y}^{\text{'}}=-3y+q\left(x,y\right)$, where $p$ and $q$ are homogeneous polynomials of degree three (either of which may be zero), that possess a first integral in a neighborhood of (0,0) of the form ${x}^{3}y+\cdots$, where omitted terms are of order at least five. Such systems are called $1:-3$ resonant centers.
##### MSC:
 34C05 Location of integral curves, singular points, limit cycles (ODE) 34C20 Transformation and reduction of ODE and systems, normal forms
##### References:
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