  We study the impact of the non-analytic reconstruction of vortex cores on
static vortex structures in weakly coupled superfluids. We show that in
rotating two-dimensional systems, the Abrikosov vortex lattice is unstable to
vortex core deformation: Each zero of the wave function becomes a cut of finite
length. The directors characterizing the orientations of the cuts are
themselves ordered in superstructures due either to surface effects or to
interaction with shear deformations of the lattice (spiral structure). Similar
instability may be also observable in clean superconducting films.
