Abstract: Based on the canonical symmetries of constrained Hamilton system, the extended canonical Nother identical equations decided by the extended Hamiltonian are formulated. It has been shown that the constraint (Lagrange) multipliers associated with the first-class constraints may not be arbitrary whether along the moving path of system or away from the path. This implies that Dirac's conjecture is questioned. A new counterexample is given for disscussion on the invalidity of Dirac's conjesture in the example. But the discussions involve no linearization of constraint.