This paper deals with the dynamic stability of autonomous weakly damped symmetric (potential) systems. Conditions for the occurrence of a limit cycle mode of instability are established via a through discussion of the effect of the damping matrix on the Jacobian eigenvalues. It was found that such a response may occur through a new type of local dynamic bifurcation identified as an isolated Ilopf bifurcation as well as through a double zero (eigenvalue) local dynamic bifurcation. As a consequence of this, undamped stable symmetric systems may become unstable with the inclusion of damping. Numerical results confirm the validity of the theoretical findings presented herein.