  The selection of Saffman-Taylor fingers by surface tension has been
extensively investigated. In this paper we are concerned with the existence and
selection of steadily translating symmetric finger solutions in a Hele-Shaw
cell by small but non-zero kinetic undercooling ($\epsilon^2 $). We rigorously
conclude that for relative finger width $\lambda$ near one half, symmetric
finger solutions exist in the asymptotic limit of undercooling $\epsilon^2
~\rightarrow ~0$ if the Stokes multiplier for a relatively simple nonlinear
differential equation is zero. This Stokes multiplier $S$ depends on the
parameter $\alpha \equiv \frac{2 \lambda
-1}{(1-\lambda)}\epsilon^{-\frac{4}{3}} $ and earlier calculations have shown
this to be zero for a discrete set of values of $\alpha$. While this result is
similar to that obtained previously for Saffman-Taylor fingers by surface
tension, the analysis for the problem with kinetic undercooling exhibits a
number of subtleties as pointed out by Chapman and King (2003) [The selection
of Saffman-Taylor fingers by kinetic undercooling, Journal of Engineering
Mathematics 46, 1-32]. The main subtlety is the behavior of the Stokes lines at
the finger tip, where the analysis is complicated by non-analyticity of
coefficients in the governing equation.
