By　utilizing　a　neural-network-based　adaptive　critic　mechanism,　the　optimal　tracking　control　problem　is　investigated　for　nonlinear　continuous-time　(CT)　multiplayer　zero-sum　games　(ZSGs)　with　asymmetric　constraints.　Initially,　we　build　an　augmented　system　with　the　tracking　error　system　and　the　reference　system.　Moreover,　a　novel　nonquadratic　function　is　introduced　to　address　asymmetric　constraints.　Then,　we　derive　the　tracking　Hamilton-Jacobi-Isaacs　(HJI)　equation　of　the　constrained　nonlinear　multiplayer　ZSG.　However,　it　is　extremely　hard　to　get　the　analytical　solution　to　the　HJI　equation.　Hence,　an　adaptive　critic　mechanism　based　on　neural　networks　is　established　to　estimate　the　optimal　cost　function,　so　as　to　obtain　the　near-optimal　control　policy　set　and　the　near　worst　disturbance　policy　set.　In　the　process　of　neural　critic　learning,　we　only　utilize　one　critic　neural　network　and　develop　a　new　weight　updating　rule.　After　that,　by　using　the　Lyapunov　approach,　the　uniform　ultimate　boundedness　stability　of　the　tracking　error　in　the　augmented　system　and　the　weight　estimation　error　of　the　critic　network　is　verified.　Finally,　two　simulation　examples　are　provided　to　demonstrate　the　efficacy　of　the　established　mechanism.