Summary: We study a modified Stefan problem (and its quasi-steady approximation) for crystalline motion in the plane. We are interested in the behaviour of the solution to a symmetric problem, in particular we assume that the Wulff shape \(W\) is a regular polygon with \(N\) sides. We describe two situations. In the first one we show that ice will be melting. In the second one we examine the properties of \(V(t)\) for small \(t\) assuming that \(V(0)=0\), where \(V\) is the velocity of the interfacial curve.