  A new approach for solving the Bethe ansatz (Gaudin-Richardson) equations of
the standard pairing problem is established based on the Heine-Stieltjes
correspondence. For $k$ pairs of valence nucleons on $n$ different
single-particle levels, it is found that solutions of the Bethe ansatz
equations can be obtained from one (k+1)x(k+1) and one (n-1)x(k+1) matrices,
which are associated with the extended Heine-Stieltjes and Van Vleck
polynomials, respectively. Since the coefficients in these polynomials are free
from divergence with variations in contrast to the original Bethe ansatz
equations, the approach thus provides with a new efficient and systematic way
to solve the problem, which, by extension, can also be used to solve a large
class of Gaudin-type quantum many-body problems and to establish a new
efficient angular momentum projection method for multi-particle systems.
