Using a model of local non-equilibrium diffusion during rapid solidification of a binary system, the isosolutal shapes of growing crystals in steady-state approximation are obtained. It is found that for crystals growing with constant velocity along a selected coordinate direction, two isosolutal growth shapes can occur. These are: the parabolic platelet in two-dimensional case and the paraboloid of revolution in three-dimensional case. In the isothermal case of diffusionless solidification, when the velocity of solidification is equal to or greater than the solute diffusive speed in the bulk system, these interfaces can have an arbitrary configuration. Special attention is given to mathematical transformations from parabolic (paraboloidal) coordinates to usual Cartesian coordinates for Ivantsov solutions extended to the case of rapid dendritic growth in which the solidification velocity V is comparable with the solute diffusion speed V-D in bulk liquid. (C) 2015 Elsevier Ltd. All rights reserved.