Let QSH be the family of non-degenerate planar quadratic differential systems possessing an invariant hyperbola. We study this class from the viewpoint ofintegrability. This is a rich family with a variety of integrable systems with either polynomial, rational, Darboux or more general Liouvillian first integrals as well as nonintegrable systems. We are interested in studying the integrable systems in this familyfrom the topological, dynamical and algebraic geometric viewpoints. In this work weperform this study for three of the normal forms of QSH, construct their topologicalbifurcation diagrams as well as the bifurcation diagrams of their configurations of invariant hyperbolas and lines and point out the relationship between them. We showthat all systems in one of the three families have a rational first integral. For anotherone of the three families, we give a global answer to the problem of Poincaré by producing a geometric necessary and sufficient condition for a system in this family to havea rational first integral. Our analysis led us to raise some questions in the last section,relating the geometry of the invariant algebraic curves (lines and hyperbolas) in thesystems and the expression of the corresponding integrating factors.
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