  For the class of anisotropic Kepler problems in $\RR^d\setminus\{0\}$ with
homogeneous potentials, we seek parabolic trajectories having prescribed
asymptotic directions at infinity and which, in addition, are Morse minimizing
geodesics for the Jacobi metric. Such trajectories correspond to saddle
heteroclinics on the collision manifold, are structurally unstable and appear
only for a codimension-one submanifold of such potentials. We give them a
variational characterization in terms of the behavior of the parameter-free
minimizers of an associated obstacle problem. We then give a full
characterization of such a codimension-one manifold of potentials and we show
how to parameterize it with respect to the degree of homogeneity.
