An ice sheet will, in general, possess points where the horizontal component of velocity is zero, and some of these will be ice centres, occurring close to summits. The paper examines the possible flow patterns near such points. The corresponding horizontal strain-rate pattern is studied by considering an ice sheet which initially has perfect circular symmetry about a vertical axis. Before perturbation there is an isotropic point for the horizontal surface strain rate at the centre. It may be shown, on purely topological grounds and without any reference to the mechanism of flow, that, when the symmetry is broken, this point, being degenerate and structurally unstable, breaks up into two structurally stable components. The breakup always occurs in essentially the same way. Around the two component points the trajectories of principal strain-rate directions always have the lemon pattern. The contours of equal principal strain rate around them are usually hyperbolic; however, if the unperturbed flow pattern had a very pronounced spiral character, they would be elliptic. This behaviour is in contrast to that of the ice centre itself, which remains unsplit.