  The Tolman-Oppenheimer-Volkov [TOV] equation constrains the internal
structure of general relativistic static perfect fluid spheres. We develop
several "solution generating" theorems for the TOV, whereby any given solution
can be "deformed" to a new solution. Because the theorems we develop work
directly in terms of the physical observables -- pressure profile and density
profile -- it is relatively easy to check the density and pressure profiles for
physical reasonableness. This work complements our previous article [Phys. Rev.
D71 (2005) 124307; gr-qc/0503007] wherein a similar "algorithmic" analysis of
the general relativistic static perfect fluid sphere was presented in terms of
the spacetime geometry -- in the present analysis the pressure and density are
primary and the spacetime geometry is secondary. In particular, our "deformed"
solutions to the TOV equation are conveniently parameterized in terms of delta
rho_c and delta p_c, the finite shift in the central density and central
pressure. We conclude by presenting a new physical and mathematical
interpretation of the TOV equation -- as an integrability condition on the
density and pressure profiles.
