The paper offers the complete form of the BRST charges and of the extended
Hamiltonian for the first-rank theories in the context of the sp(3) BRST sy
mmetry. A theorem is proposed in order to solve the problem of existence an
d uniqueness of a gauge fixed Hamiltonian in this context. As a general res
ult, it is verified that the standard and the sp(2) theories could be recov
ered as particular cases of our generalized symmetry.