CANONICAL FORMS OF CUBIC DIFFERENTIAL SYSTEMS WITH REAL INVARIANT STRAIGHT LINES OF TOTAL MULTIPLICITY SEVEN ALONG ONE DIRECTION

Abstract

Consider the general cubic differential system x  Px, y , y  Qx, y , where
P Q x y , , R , max deg ,deg 3  P Q  , GCD P Q  , 1   . If this system has enough invariant straight
lines considered with their multiplicities, then, according to [1], we can construct a Darboux first integral.
In this paper we obtain 26 canonical forms for cubic differential systems which possess real invariant
straight lines along one direction of total multiplicity seven including the straight line at the infinity.