Planar quadratic differential systems with invariant straight lines of total multiplicity four

In this article we prove a classification theorem of real planar quadratic vector fields which possess four invariant straight lines, including the line at infinity and including their multiplicities. This classification, which is taken modulo the action of the group of real affine transformations and time rescaling, is given in terms of algebraic invariants and comitants and also geometrically, using cycles on the complex projective plane and on the line at infinity. The algebraic invariants and comitants allow us to verify for any given real quadratic system whether or not it has invariant lines of total multiplicity four, and to specify its configuration of lines endowed with their corresponding real singularities of this system. The calculations can be implemented on computer. This classification is instrumental in studying the problem of Darboux integrability of such systems, work which is in progress.