  The Gutzwiller's trace formula for the anisotropic Kepler problem is Fourier
transformed with a convenient variable $u=1/\sqrt{-2E}$ which takes care of the
scaling property of the AKP action $S(E)$. Proper symmetrization procedure
(Gutzwiller's prescription) is taken by the introduction of half-orbits that
close under symmetry transformations so that the two dimensional semi-classical
formulas match correctly the quantum subsectors $m^\pi=0^+$ and $m^\pi=0^-$.
Response functions constructed from half orbits in the POT side are explicitly
given. In particular the response function $g_X$ from $X$-symmetric half orbit
has an amplitude where the root of the monodromy determinant is inverse
hyperbolic.
  The resultant {\it weighted densities of periodic orbits $D^{m=0}_e(\phi)$
and $D^{m=0}_o(\phi)$} from both quantum subsectors show peaks at the actions
of the periodic orbits with correct peak heights corresponding to the Lyapunov
exponents of them. The formulation takes care of the cut off of the energy
levels, and the agreement between the $D(\phi)$s of QM and POT sides is
observed independent for the choice of the cut off. The systematics appeared in
the densities of the periodic orbits is explained in terms of features of
periodic orbits.
  It is shown that from quantum energy levels one can extract the information
of AKP periodic orbits, even the Lyapunov exponents-- the success of inverse
quantum chaology in AKP.
