  In this paper we revisit the problem of finding hidden symmetries in quantum
mechanical systems. Our interest in this problem was renewed by nontrivial
degeneracies of a simple spin Hamiltonian used to model spin relaxation in
alkali-metal vapors. We consider this spin Hamiltonian in detail and use this
example to outline a general approach to finding symmetries when eigenvalues
and eigenstates of the Hamiltonian are known. We extract all nontrivial
symmetries responsible for the degeneracy and show that the symmetry group of
the Hamiltonian is SU(2). The symmetry operators have a simple meaning which
becomes transparent in the limit of large spin. As an additional example we
apply the method to the Hydrogen atom.
