The one-sided model for a nearly planar solidification front advancing at steady velocity is studied. The model neglects impurity diffusion in the solid; the interface is stabilized by the imposition of a thermal gradient. The front, located at ${z}_{s}={z}_{s}(x,y)$, is described by the coefficients ${\ensuremath{\epsilon}}_{\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}}$ of the Fourier expansion for ${z}_{s}$, and equations of motion ${\ensuremath{\epsilon}}_{\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}}={f}_{\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}}({\ensuremath{\epsilon}})$ are derived in the approximation where the velocity $\stackrel{\ensuremath{\rightarrow}}{\mathrm{v}}$ of the interface is small. The functions ${f}_{\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}}$ are expressed as infinite polynomials in the ${\ensuremath{\epsilon}}$. Stationary solutions ${f}_{\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}}=0$ are sought with the help of a consistent truncation scheme. Truncations which involve keeping terms of up to fifth order in ${\ensuremath{\epsilon}}_{\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}}$ are used. Stationary profiles for both one- and two-dimensional fronts are obtained numerically, whose features are in reasonable agreement with experimental observations. In particular, the one-dimensional solutions exhibit a relatively well-developed cellular structure; this is in contrast with what happens in more conventional analyses, where higher-order nonlinearities are not accounted for. The two-dimensional stationary interfaces are of various types, displaying twofold or sixfold symmetry. It appears to be the first time that calculations of two-dimensional structures are reported.