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The Tolman-Oppenheimer-Volkov [TOV] equation constrains the internalstructure of general relativistic static perfect fluid spheres. We developseveral "solution generating" theorems for the TOV, whereby any given solutioncan be "deformed" to a new solution. Because the theorems we develop workdirectly in terms of the physical observables -- pressure profile and densityprofile -- it is relatively easy to check the density and pressure profiles forphysical reasonableness. This work complements our previous article [Phys. Rev.D71 (2005) 124307; gr-qc/0503007] wherein a similar "algorithmic" analysis ofthe general relativistic static perfect fluid sphere was presented in terms ofthe spacetime geometry -- in the present analysis the pressure and density areprimary and the spacetime geometry is secondary. In particular, our "deformed"solutions to the TOV equation are conveniently parameterized in terms of deltarho_c and delta p_c, the finite shift in the central density and centralpressure. We conclude by presenting a new physical and mathematicalinterpretation of the TOV equation -- as an integrability condition on thedensity and pressure profiles.Comment: V1: 11 pages; uses iopart.sty; V2: 11 pages, uses revetx4.sty,  significant additional material. This version accepted for publication in  Physical Review

Solution generating theorems for the Tolman-Oppenheimer-Volkov equation