- This paper addresses the continuum models of solidification of a monocomponent substance, that generalize the classical Stefan problem by incorporating effects of interfacial kinetics. We consider the key planar solidification problem of this sort. The model implies a rigid solid and an incompressible liquid of equal densities and accounts for a linear attachment kinetics at the interface. The correctness of this formulation is proved and the asymptotic solutions at the onset of freezing and at sufficiently long times are developed. An analytical solution, valid uniformly in time, is developed and validated against the finite difference solution.