  We review the solution space for the field equations of Einstein's General
Relativity for various static, spherically symmetric spacetimes. We consider
the vacuum case, represented by the Schwarzschild black hole; the de
Sitter-Schwarzschild geometry, which includes a cosmological constant; the
Reissner-Nordstr\"{o}m geometry, which accounts for the presence of charge.
Additionally we consider the homogenenous and anisotropic locally rotational
Bianchi II spacetime in the vacuum. Our analysis reveals that the field
equations for these scenarios share a common three-dimensional group of point
transformations, with the generators being the elements of the
$D\otimes_{s}T_{2}$ Lie algebra, known as the semidirect product of dilations
and translations in the plane. Due to this algebraic property the field
equations for the aforementioned gravitational models can be expressed in the
equivalent form of the null geodesic equations for conformally flat geometries.
Consequently, the solution space for the field equations is common, and it is
the solution space for the free particle in a flat space. This appoach open new
directions on the construction of analytic solutions in gravitational physics
and cosmology.
