  We focus on existence and rigidity problems of the vectorial Peierls-Nabarro
(PN) model for dislocations. Under the assumption that the misfit potential on
the slip plane only depends on the shear displacement along the Burgers vector,
a reduced non-local scalar Ginzburg-Landau equation with an anisotropic
positive (if Poisson ratio belongs to $(-1/2,1/3)$) singular kernel is derived
on the slip plane. We first prove that minimizers of the PN energy for this
reduced scalar problem exist. Starting from $H^{1/2}$ regularity, we prove that
these minimizers are smooth 1D profiles only depending on the shear direction,
monotonically and uniformly converge to two stable states at far fields in the
direction of the Burgers vector. Then a De Giorgi-type conjecture of
single-variable symmetry for both minimizers and layer solutions is
established. As a direct corollary, minimizers and layer solutions are unique
up to translations. The proof of this De Giorgi-type conjecture relies on a
delicate spectral analysis which is especially powerful for nonlocal
pseudo-differential operators with strong maximal principle. All these results
hold in any dimension since we work on the domain periodic in the transverse
directions of the slip plane. The physical interpretation of this rigidity
result is that the equilibrium dislocation on the slip plane only admits shear
displacements and is a strictly monotonic 1D profile provided exclusive
dependence of the misfit potential on the shear displacement.
