The occurrence of generic degeneracies in physical systems is closely related to underlying symmetries of the governing equations. The occurrence of additional non-generic degeneracies which cannot be accounted for by symmetry arguments is usually termed as accidental. In this work, we formulate a mechanistic framework which helps identify and investigate a particular class of degeneracies associated with equivariant systems under certain common symmetry groups. We show that the existence of a first- integral for such systems (i.e., a potential function or energy functional) guarantees generically that non-generic degeneracies in the spectrum of the Jacobian (and likely other properties of the system) occurs. We apply our theory to some common physical systems and show that it successfully explains the "accidental" degeneracy found in (1) periodic crystals without inversion symmetry, (2) quantum mechanical calculations involving cyclic nano-structures and (3) the well-know degeneracies of the elastic constants matrix in the theory of linear elasticity.