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For static fluid spheres, the condition of hydrostatic equilibrium is givenby the generalized Tolman--Oppenheimer--Volkoff (TOV) equation, a Riccatiequation in the radial pressure. For a perfect fluid source, it is known thatfinding a new solution from an existing solution requires solving a Bernoulliequation, if the density profile is kept the same. In this paper, we considermaps between static (an)isotropic fluid spheres with the same (arbitrary)density profile and present solution-generating techniques to find newsolutions from existing ones. The maps, in general, require solving anassociated Riccati equation, which, unlike the Bernoulli equation, cannot besolved by quadrature. In any case, it can be shown that the output solution isnot, in general, regular for a given regular input solution. However, ifpressure anisotropy is kept the same, the new solution is both regular and canbe found by solving a Bernoulli equation. We give a few examples where thegeneralized TOV equation, under algebraic constraints, can be converted into aBernoulli equation and thus, solved exactly. We discuss the physicalsignificance of these Bernoulli equations. Since the density profile remainsthe same in our approach, the spatial line element is identical for allsolutions, which facilitates direct comparison between various equilibriumconfigurations using fluid variables as functions of the radial coordinate.Finally, combining with the previous study on generation algorithms, we showhow this study leads us to a new three-parameter family of exact solutions thatsatisfy all desirable physical conditions.

Generating Spherically Symmetric Static Anisotropic Fluid Solutions of Einstein's Equations from Hydrostatic Equilibrium