Centers of projective vector fields of spatial quasi-homogeneous systems with weight (m,m, n) and degree 2 on the sphere

© 2016, University of Szeged. All rights reserved. In this paper we study the centers of projective vector fields QT of threedimensional quasi-homogeneous differential system dx/dt = Q(x) with the weight (m,m, n) and degree 2 on the unit sphere S2. We seek the sufficient and necessary conditions under which QT has at least one center on S2. Moreover, we provide the exact number and the positions of the centers of QT. First we give the complete classification of systems dx/dt = Q(x) and then, using the induced systems of QT on the local charts of S2, we determine the conditions for the existence of centers. The results of this paper provide a convenient criterion to find out all the centers of QT on S2 with Q being the quasi-homogeneous polynomial vector field of weight (m,m, n) and degree 2.