In　this　paper,　we　study　the　Riemann　problem　with　the　initial　data　containing　the　Dirac　delta　function　for　the　nonsymmetric　Keyfitz-Kranzer　system　with　Chaplygin　pressure.　Under　the　generalized　Rankine-Hugoniot　conditions　and　entropy　condition,　we　constructively　obtain　the　global　existence　of　generalized　solutions　including　delta　shock　waves　that　explicitly　exhibit　four　kinds　of　different　structures.　Moreover,　we　obtain　the　stability　of　generalized　solutions　by　making　use　of　the　perturbation　of　the　initial　data.