Direct product of unfurled swallowtails in the problem of going around an obstacle (Q1364866)

Lagrangian varieties arising in the problem of going around an obstacle bounded by a smooth hypersurface \(\Gamma\) in a smooth configuration manifold are studied. Let, for example, \(\Gamma\) be a smooth hypersurface in Euclidean space. Consider rays starting from a given point and touching \(\Gamma\). They define the variety \(l\) of geodesics on \(\Gamma\) starting from contact points of rays with \(\Gamma\). Let \(L\) be a variety of rays in the ambient space which break loose from the pencil of geodesics given by the variety \(l\). The author gives formal normal forms up to symplectomorphisms (diffeomorphisms) of a generic variety \(L\) in a neighbourhood of rays 2-tangent (3-tangent) to \(\Gamma\).