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We discuss a hierarchy of broken symmetries with special emphasis on partialdynamical symmetries (PDS). The latter correspond to a situation in which anon-invariant Hamiltonian accommodates a subset of solvable eigenstates withgood symmetry, while other eigenstates are mixed. We present an algorithm forconstructing Hamiltonians with this property and demonstrate the relevance ofthe PDS notion to nuclear spectroscopy, to quantum phase transitions and tomixed systems with coexisting regularity and chaos.

Exact Dynamical and Partial Symmetries