Few ice sheet flow models have been developed that solve the complete set of mechanical equations. Until now, these models were limited to isotropic conditions. We present here a two-dimensional, finite difference method capable of solving the equations for the steady flow of a viscous, incompressible, anisotropic fluid with a free surface under isothermal conditions. It is not a standard method, especially with respect to the time discretization of the numerical scheme, and converges for very low Reynolds numbers. This method is applied here to the planar flow of anisotropic ice over flat or irregular bedrock, with no-slip boundary conditions at the ice-bedrock interface. The results are presented here for Newtonian behavior in the vicinity of an ice divide. The ice is assumed to be isotropic at the ice sheet surface, with continuous and prescribed development of anisotropy with increasing depth. Going from isotropic to anisotropic situations, our results indicate that the free surface becomes flatter and the shear strain rates larger and more concentrated near the bedrock. The flow is less sensitive to variations of the bedrock topography in the anisotropic case than in the isotropic case. Furthermore, a new phenomenon appears in the anisotropic case: the partial stagnation of ice in the holes of the bedrock. These effects have significant consequences when dating the ice. The isochrones obtained in the anisotropic case are flatter and the anisotropic ice is more than 10% younger above the bumps and more than 100% older within the holes than for the isotropic ice.¿ 1997 American Geophysical Union