  We study static spherically symmetric anisotropic fluid solutions of the
Einstein equations in curvature coordinates to find algorithms that generate
all solutions of the system and sub-algorithms that generate all solutions that
are regular at the center. For this, we study all possible combinations of
input functions from the set of four functions that characterize the
anisotropic system and revisit the conditions for central regularity of fluid
spheres (isotropic and anisotropic) to see how different sets of initial
conditions serve as equivalent sets for regularity. We provide the first
regularity analysis of the solutions generated by the well-known algorithm that
uses the potential function and anisotropy. This includes the algorithm for the
isotropic system as a special case. We show that the one-parameter family of
solutions produced by either algorithm are regular at the center for the usual
initial conditions for the input functions; these conditions by themselves are
inadequate for geometric regularity, but the Einstein equations fill the gap.
By reformulating the anisotropic algorithm in terms of the potential function
and tangential pressure, all regular anisotropic solutions can further be
obtained in a 1-1 fashion with the input. This allows us to interpret the
parameter in the existing algorithms for the isotropic and anisotropic systems
as the central density. For three other choices of input function pairs (any
two of the three functions: the potential function, density, and radial
pressure), a remarkably straightforward algorithm follows. The equivalency of
the three pairs in this algorithm arises precisely because of the same
algebraic relation between these three functions that made the different
equivalent sets of regularity conditions possible. This makes this algorithm
very streamlined for generating regular anisotropic solutions.
