The projective vector field of a kind of three-dimensional quasi-homogeneous system on S2S2

In this paper we study the projective vector field QT of a three-dimensional quasi-homogeneous system with weight (1,1,α3)(1,1,α3) and degree δ=2δ=2, α3≥2α3≥2. Projective vector fields QT of this kind are classified into two types. For one type, QT has no closed orbit and at most eight singularities, which lead to a global topology of the three-dimensional system. For the other type, QT has at most ten singularities. In addition, we show a relationship between QT and a Lienard system of this type. For both of them we obtain some conditions for the existence of limit cycles.