We study the existence and structure of steady-state ngers in two-dimensional solidi cation, when the surface energy has a crystalline anisotropy so that the energy-minimizing Wul shape and hence the solid-liquid interface are polygons, and in the one-sided quasi-static limit so that the di usion eld satis es Laplace's equation in the liquid. In a channel of nite width, this problem is the crystalline analog of the classic Sa man-Taylor smooth nger in Hele-Shaw ow. By a combination of analysis

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... nd numerical Schwarz-Christo el mapping methods, we show that, as for solutions of the smooth problem, for each choice of Wul shape there is a critical maximum value of the magnitude of surface tension above which no convex steady-state solutions exist. We then exhibit convergence of convex crystalline solutions to convex smooth solutions as the Wul shape approaches a circle. We also consider the open dendrite geometry, and show that there are no steady-state solutions having a nite number of sides for any crystalline surface energy. This is in striking contrast to the smooth case, and is an indication that the time-dependent behavior may be more complicated for crystalline surface energies.