Singular improper affine spheres from a given Lagrangian submanifold

Given a Lagrangian submanifold L of the affine symplectic 2n-space, one can canonically and uniquely define a center-chord and a special improper affine sphere of dimension 2n, both of whose sets of singularities contain L. Although these improper affine spheres (IAS) always present other singularities away from L (the off-shell singularities studied in [8]), they may also present singularities other than L which are arbitrarily close to L, the so called singularities “on shell”. These on-shell singularities possess a hidden Z2 symmetry that is absent from the off-shell singularities. In this paper, we study these canonical IAS obtained from L and their on-shell singularities, in arbitrary even dimensions, and classify all stable Lagrangian/Legendrian singularities on shell that may occur for these IAS when L is a curve or a Lagrangian surface.