This　paper　presents　new　results　on　the　limit　cycles　of　a　Liénard　system　with　symmetry　allowing　for　discontinuity.　Our　results　generalize　and　improve　the　results　in　[32,　Theorem　1　and　2]　or　the　monograph　[33,　Chapter　4,　Theorem　5.2].　The　results　in　[33]　are　only　valid　for　the　smooth　system.　We　emphasize　that　our　main　results　are　valid　for　discontinuous　systems.　Moreover,　we　show　the　presence　and　an　explicit　upper　bound　for　the　amplitude　of　the　two　limit　cycles,　and　we　estimate　the　position　of　the　double-limit-cycle　bifurcation　surface　in　the　parameter　space.　Until　now,　there　is　no　result　to　determine　the　amplitude　of　the　two　limit　cycles.　The　existing　results　on　the　amplitude　of　limit　cycles　guarantee　that　the　Liénard　system　has　a　unique　limit　cycle.　Finally,　some　applications　and　examples　are　provided　to　show　the　effectiveness　of　our　results.　We　revisit　a　co-dimension-3　Liénard　oscillator　(see　[22,34])　in　Application　1.　Li　and　Rousseau　[22]　studied　the　limit　cycles　of　such　a　system　when　the　parameters　are　small.　However,　for　the　general　case　of　the　parameters　(in　particular,　the　parameters　are　large),　the　upper　bound　of　the　limit　cycles　remains　open.　We　completely　provide　the　bifurcation　diagram　for　the　one-equilibrium　case.　Moreover,　we　determine　the　amplitude　of　the　two　limit　cycles　and　estimate　the　position　of　the　double-limit-cycle　bifurcation　surface　for　the　one-equilibrium　case.　Application　2　is　presented　to　study　the　limit　cycles　of　a　class　of　the　Filippov　system.　©　2018　Elsevier　Inc.