The　gauge　symmetries　of　a　constrained　system　can　be　deduced　from　the　gauge　identities　with　Lagrange　method,　or　the　first-class　constraints　with　Hamilton　approach.　If　Dirac　conjecture　is　valid　to　a　dynamic　system,　in　which　all　the　first-class　constraints　are　the　generators　of　the　gauge　transformations,　the　gauge　transformations　deduced　from　the　gauge　identities　are　consistent　with　these　given　by　the　first-class　constraints.　Once　the　equivalence　vanishes　to　a　constrained　system,　in　which　Dirac　conjecture　would　be　invalid.　By　using　the　equivalence,　two　counterexamples　and　one　example　to　Dirac　conjecture　are　discussed　to　obtain　defined　results.