Electronic Journal of Differential Equations, Vol. 2021 (2021), No. 69, pp. 1-52.

Title: Quadratic systems with an invariant algebraic curve of degree 3 and a Darboux invariant
       
Authors: Jaume Llibre (Univ.Autonoma de Barcelona, Catalonia, Spain)
         Regilene Oliveira (Univ. de Sao Paulo, SP, Brazil)
         Camila Rodrigues (Univ. Federal de Santa Catarina, Brazil)

Abstract:
 Let QS be the class of non-degenerate planar quadratic differential systems
 and  QS<sub>3</sub> its subclass formed by the systems possessing an invariant 
 cubic f(x,y)=0.  In this article, using the action of the group of real affine 
 transformations and time  rescaling on QS, we obtain all the possible normal 
 forms for the quadratic  systems in QS<sub>3</sub>. 
 Working with these normal forms we complete the characterization of
 the phase portraits in QS<sub>3</sub> having a Darboux invariant of the form 
 f(x, y)e<sup>st</sup>,  with s in R.

Submitted September 25, 2020. Published August 16, 2021.
Math Subject Classifications: 34C05, 34A34, 34C23.
Key Words: Quadratic vector fields; algebraic invariant curve; Darboux invariant; 
           global phase portrait.