In　this　paper,　we　study　the　Riemann　problem　with　the　initial　data　containing　the　Dirac　delta　function　in　both　components　for　a　nonstrictly　hyperbolic　system　of　two　conservation　laws　from　zero-pressure　gas　dynamics　by　introducing　the　new　state　variabe.　In　solutions,　we　find　another　kind　of　delta　shocks　on　which　both　state　variables　simultaneously　contain　the　Dirac　delta　functions.　It　is　quite　different　from　the　previous　ones　on　which　only　one　state　variable　contains　the　Dirac　delta　function.　This　phenomenon　is　also　captured　numerically　and　experimentally　by　Kreji,　Kruni　and　Nedeljkov　[N.　Kreji,　T.　Kruni,　M.　Nedeljkov,　Numerical　verification　of　delta　shock　waves　for　pressureless　gas　dynamics,　J.　Math.　Anal.　Appl.　345　(2008)　243-257].　Moreover,　a　new　kind　of　nonclassical　wave,　namely　a　delta　contact　discontinuity,　is　discovered　here,　which　is　a　Dirac　delta　function　supported　on　a　contact　discontinuity.