| Summary: | In this paper we study the centers of projective vector fields $\mathbf{Q}_T$ of three-dimensional quasi-homogeneous differential system $d\mathbf{x}/dt=\mathbf{Q}(\mathbf{x})$ with the weight $(m,m,n)$ and degree $2$ on the unit sphere $\mathbb{S}^2$. We seek the sufficient and necessary conditions under which $\mathbf{Q}_T$ has at least one center on $\mathbb{S}^2$. Moreover, we provide the exact number and the positions of the centers of $\mathbf{Q}_T$. First we give the complete classification of systems $d\mathbf{x}/dt=\mathbf{Q}(\mathbf{x})$ and then, using the induced systems of $\mathbf{Q}_T$ on the local charts of $\mathbb{S}^2,$ 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 $\mathbf{Q}_T$ on $\mathbb{S}^2$ with $\mathbf{Q}$ being the quasi-homogeneous polynomial vector field of weight $(m,m,n)$ and degree $2$.
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