The projective vector field of a kind of three-dimensional quasi-homogeneous system on S2

In this paper we study the projective vector field View the MathML sourceQT 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 View the MathML sourceQT of this kind are classified into two types. For one type, View the MathML sourceQT has no closed orbit and at most eight singularities, which lead to a global topology of the three-dimensional system. For the other type, View the MathML sourceQT has at most ten singularities. In addition, we show a relationship between View the MathML sourceQT and a Lienard system of this type. For both of them we obtain some conditions for the existence of limit cycles.