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In the variational study of singular Lagrange systems, the zero energy solutions play an important role. The anisotropic Kepler problem is such a singular system introduced by physicist M. Gutzwiller to reveal connections between classic and quantum chaos. In this paper we find a simple way of computing the Morse indices of the zero solutions in this problem. In particular, we show the Morse indices are connected with the oscillating behaviors of these solutions. Our results can also be applied to some problems in celestial mechanics.

An Index Theory for Zero Energy Solutions of the Planar Anisotropic Kepler Problem