Elliptical, hyperbolic and parabolic dislocations

The paper discusses dislocations in an elastic medium within the geometric theory of defects. All locally flat separable metrics in three dimensions are found. They describe dislocations that occur by cutting (inserting) a wedge, ellipsoid, hyperboloid or paraboloid of rotation and elliptical, hyperbolic or parabolic cylinders with subsequent gluing of the formed edges. The obtained dislocations have an important property: variables in the corresponding Hamilton–Jacobi equation for geodesics are completely separable resulting in the Liouville integrability of geodesic equations. In addition, variables in the Laplace equation also admit complete separation of variables. To our knowledge, elliptic, hyperbolic and parabolic dislocations are described for the first time.