  We study the minimal class of exact solutions of the Saffman-Taylor problem
with zero surface tension, which contains the physical fixed points of the
regularized (non-zero surface tension) problem. New fixed points are found and
the basin of attraction of the Saffman-Taylor finger is determined within that
class. Specific features of the physics of finger competition are identified
and quantitatively defined, which are absent in the zero surface tension case.
This has dramatic consequences for the long-time asymptotics, revealing a
fundamental role of surface tension in the dynamics of the problem. A
multifinger extension of microscopic solvability theory is proposed to
elucidate the interplay between finger widths, screening and surface tension.
