Topological phase portraits of planar semi-linear quadratic vector fields

In this paper we solve completely the topological classification of the phase portraits for a class of semi-linear quadratic vector fields, i.e. the vector fields of the form X = (ax + by, Ax + By + Cx2 + Dxy + Ey2). As a corollary of our results we answer the problem proposed by Ye Yanqian at the end of §2 of [30]. Moreover, we prove that quadratic systems of class (I) in the Chinese classification of quadratic systems have exactly 50 different topological phase portraits, which corrects the result that such quadratic systems have only 47 different topological phase portraits (see Theorem 12.3 of Ye [30]).