We　study　a　family　of　quadratic　vector　fields　in　Class　Ix˙　=　y,　y˙　=　-　x-　αy+　μx2-　y2,　where　(α,　μ)　∈　R2.　To　study　the　equilibria　at　infinity　on　the　Poincaré　disk　of　this　system　completely,　we　follow　the　method　of　generalized　normal　sectors　of　Tang　and　Zhang　(Nonlinearity　17:1407–1426,　2004)　and　give　further　two　new　criterions,　which　allows　us　to　obtain　not　only　the　qualitative　properties　of　the　equilibria　but　also　asymptotic　expressions　of　these　orbits　connecting　the　equilibria　at　infinity　of　this　system.　Further,　the　complete　bifurcation　diagram　including　saddle　connection　bifurcation　curves　of　this　system　is　given.　Moreover,　by　qualitative　properties　of　the　equilibria,　the　nonexistence　of　limit　cycle　and　rotated　properties　about　α　and　μ,　all　global　phase　portraits　on　the　Poincaré　disk　of　this　system　are　also　obtained　and　the　number　is　19.　©　2020,　Springer　Nature　Switzerland　AG.