  We study the Einstein equations of the static spherically symmetric
anisotropic fluid system in curvature coordinates to find algorithms that
generate all solutions and all solutions that are regular at the center. All
possible combinations of input functions from the set of four functions that
characterize the anisotropic system are considered and all equivalent
conditions for central regularity are determined (for both isotropic and
anisotropic systems). We provide the first regularity analysis of the known
algorithm that uses the potential function and anisotropy as inputs. For three
other choices of input function pairs (any two of the potential function,
density, or radial pressure), a remarkably straightforward algorithm follows,
which is very efficient in generating regular anisotropic solutions. This is
because the equivalency of the three pairs in this algorithm arises precisely
from the same algebraic relation that made the different equivalent sets of
regularity conditions possible. In addition, the choice of functions makes this
algorithm very suitable for finding particular solutions that admit other
desirable physical properties; we construct three examples. This algorithm does
not admit an isotropic limit although all isotropic solutions are produced as
part of the anisotropic system. The remaining two choices of input function
pairs (anisotropy with the radial pressure or density) lead to the old barriers
one encounters in the isotropic system: Riccati and Abel equations. However,
with any solution generated by the new and existing algorithms, one can now
construct the general solution of the corresponding Riccati equation to obtain
a one-parameter family of geometries for each input solution. We discuss the
regularity of the resulting solutions.
