The Casimir-Polder interaction energy between a unipolarizable point atom and a unipolarizable dielectric ring has been limited, until now, to the case when the atom is confined on the axis of symmetry of the ring. We find the generalized analytical expression for any position of the atom relative to the ring in terms of complete elliptic integrals. This is aided by the construction of a class of integrals of a Jacobian elliptic function as a linear combination of complete elliptic integrals. Our expression for the interaction energy allows us to investigate the instability of the atom even for the equilibrium points which exists off the axis of symmetry.