The article describes new types of linear dislocations in an elastic medium within the framework of the geometric theory of defects. All locally flat separable metrics in three-dimensional space are found that admit two Killing vector fields and one second rank Killing tensor. The obtained metrics have important property: the variables in the corresponding Hamilton-Jacobi equation for geodesics are completely separable which leads to Liouville integrability of geodetic equations.
Katanaev et al. (Fri,) studied this question.