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We study a one-phase Stefan problem for a semi-infinite material with temperature-dependent thermal conductivity with a constant temperature or a heat flux condition of the type $-q_0/\sqrt{t}$ ($q_0>0$) at the fixed face $x=0$. We obtain in both cases sufficient conditions for data in order to have a parametric representation of the solution of the similarity type for $t\ge t_0>0$ with $t_0$ an arbitrary positive time. These explicit solutions are obtained through the unique solution of an integral equation with the time as a parameter....