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.
