  We study the exact non-singular zero-surface tension solutions of the
Saffman-Taylor problem for all times. We show that all moving logarithmic
singularities a_k(t) in the complex plane \omega = e^{i\phi}, where \phi is the
stream function, are repelled from the origin, attracted to the unit circle and
eventually coalesce. This pole evolution describes essentially all the
dynamical features of viscous fingering in the Hele-Shaw cell observed by
Saffman and Taylor [Proc. R. Soc. A 245, 312 (1958)], namely tip-splitting,
multi-finger competition, inverse cascade, and subsequent formation of a single
Saffman-Taylor finger.
