  An ADM-like Hamiltonian approach is proposed for static spherically symmetric
relativistic star configurations. For a given equation of state the entire
information about the model can be encoded in a certain 2-dimensional
minisuperspace geometry. We derive exact solutions which arise from symmetries
corresponding to linear and quadratic geodesic invariants in minisuperspace by
exploiting the relation to minisuperspace Killing tensors. A classification of
exact solutions having the full number of integrations constants is given
according to their minisuperspace symmetry properties. In particular it is
shown that Schwarzschild's exterior solution and Buchdahl's n=1 polytrope
solution correspond to minisuperspaces with a Killing vector symmetry while
Schwarzschild's interior solution, Whittaker's solution and Buchdahl's n=5
polytrope solution correspond to minisuperspaces with a second rank Killing
tensor. New solutions filling in empty slots in this classification scheme are
also given. One of these new solutions has a physically reasonable equation of
state and is a generalization of Buchdahl's n=1 polytrope model.

General Relativity and Quantum Cosmology