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