  In this article, using an infinite-dimensional equivariant Conley index, we
prove a generalization of the profitable Liapunov center theorem for symmetric
potentials. Consider the system $\ddot{q}= -\nabla U(q),$ where $U(q)$ is a
$\Gamma$-symmetric potential, where $\Gamma$ is a compact Lie group acting
linearly on $\mathbb{R}^n$. If the system possess a non-degenerate orbit of
stationary solutions $\Gamma(q_0)$ with trivial isotropy group, such that there
exists at least one positive eigenvalue of the Hessian $\nabla^2 U(q_0)$, then
in any neighbourhood of orbit $\Gamma(q_0)$ there is a periodic orbit of
solutions of the system.
