We have derived the generalized first Noether theorem for gauge-variant system with singular Lagrangian and Noether identities for variant system in phase space. The strong and week conservation laws for variant system were deduced. Some preliminary applications were given. In certain cases a variant system in canonical variables is a constrained Hamiltonian system. Based upon the symmtry properties of such system

an example was given in which Dirac's conjecture fails

in that we do not write the constraints in linearized form as Cawley and others do.