In　this　paper,　we　study　a　quintic　Liénard　system　x.　=　y,　y.　=　-(a0x　+　a1x3　+　a2x5)　-　(b0　+　b1x2)y　with　ℤ2-equivariance,　arising　from　the　complex　Ginzburg-Landau　equation.　Although　this　system　is　a　versal　unfolding　of　the　germ　x.　=　y,　y.　=　-a2x5　+　O(x6)　-　(b1x2　+　O(x3))y　near　the　origin,　it　cannot　be　changed　equivalently　into　a　near-Hamiltonian　system　for　global　variables　and　parameters　so　that　its　dynamics　cannot　be　studied　via　counting　the　isolate　zeros　of　Abelian　integrals　as　usual.　We　present　a　complete　study　of　this　system　with　a2　＜　0,　i.e.,　the　sum　of　indices　of　equilibria　is　-1,　and　show　that　this　system　exhibits　at　most　three　limit　cycles　and　a　double　center.　The　necessary　and　sufficient　conditions　are　obtained　on　the　existence　of　three　limit　cycles,　a　stable　two-saddle　heteroclinic　loop,　an　unstable　figure-eight　loop,　and　two　stable　homoclinic　loops.　A　global　bifurcation　diagram　and　the　corresponding　global　phase　portraits　in　the　Poincaré　disc　of　this　system　are　given,　including　pitchfork　bifurcation,　Hopf　bifurcation,　transcritical　bifurcation,　two-saddle　heteroclinic　loop　bifurcation,　double　limit　cycle　bifurcation,　homoclinic　bifurcation,　saddle　connection　bifurcation,　and　degenerate　Bogdanov-Takens　bifurcation.　Note　that　the　dynamics　of　this　quintic　Liénard　system　is　so　complicated　that　it　has　infinitely　many　bifurcation　surfaces　of　saddle　connection.　Copyright　©　by　SIAM.