  Conserved quantities are crucial in quantum physics. Here we discuss a
general scenario of Hamiltonians. All the Hamiltonians within this scenario
share a common conserved quantity form. For unitary parametrization processes,
the characteristic operator of this scenario is analytically provided, as well
as the corresponding quantum Fisher information (QFI). As the application of
this scenario, we focus on two classes of Hamiltonians: su(2) category and
canonical category. Several specific physical systems in these two categories
are discussed in detail. Besides, we also calculate an alternative form of QFI
in this scenario.
