  Previously a theory has been presented which extends the geometrical
structure of a real four-dimensional space-time via a field of orthonormal
tetrads with an enlarged transformation group. This new transformation group,
called the conservation group, contains the group of diffeomorphisms as a
proper subgroup and we hypothesize that it is the foundational group for
quantum geometry. The fundamental geometric object of the new geometry is the
curvature vector, C^\mu . Using the scalar Lagrangian density C^\mu C_\mu
\sqrt{-g}, field equations for the free field have been obtained which are
invariant under the conservation group. In this paper, this theory is further
extended by development of a suitable Lagrangian for a field with sources.
Spherically symmetric solutions for both the free field and the field with
sources are given. A stellar model and an external, free-field model are
developed. The theory implies that the external stress-energy tensor has
non-compact support and hence may give the geometrical foundation for dark
matter. The resulting models are compared to the internal and external
Schwarzschild models. The theory may explain the Pioneer anomaly and the corona
heating problem.
