Configurations of quadratic systems possessing three distinct infinite singularities and one or more invariant parabolas

Denote by <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mtext mathvariant="bold">QS</mml:mtext> </mml:mrow> </mml:math> the class of all non-degenerate planar quadratic differential systems and by <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mtext mathvariant="bold">QSP</mml:mtext> </mml:mrow> </mml:math> the subclass of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mtext mathvariant="bold">QS</mml:mtext> </mml:mrow> </mml:math> of all systems possessing at least one invariant parabola. In this paper we consider the subfamily of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mtext mathvariant="bold">QSP</mml:mtext> </mml:mrow> </mml:math> defined by the condition <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>η</mml:mi> <mml:mo>≠</mml:mo> <mml:mn>0</mml:mn> </mml:math> , which means the presence of three distinct infinite singularities real or complex. We denote this subfamily by <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mtext mathvariant="bold">QSP</mml:mtext> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>η</mml:mi> <mml:mo>≠</mml:mo> <mml:mn>0</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> . We investigate all possible configurations of invariant parabolas and invariant straight lines which systems in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mtext mathvariant="bold">QSP</mml:mtext> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>η</mml:mi> <mml:mo>≠</mml:mo> <mml:mn>0</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> could possess and their geometric properties encoded in such configurations. The classification presented here is taken modulo the action of the group of real affine transformations and time rescaling and it is given in terms of affine invariant polynomials. It yields a total of 146 distinct configurations. The obtained classification is an algorithm which makes it possible for any given real quadratic differential system in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mtext mathvariant="bold">QSP</mml:mtext> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>η</mml:mi> <mml:mo>≠</mml:mo> <mml:mn>0</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> to specify its configuration of invariant parabolas and straight lines. This work will prove helpful in studying the integrability of the systems in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mtext mathvariant="bold">QSP</mml:mtext> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>η</mml:mi> <mml:mo>≠</mml:mo> <mml:mn>0</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> .