  We study inhomogeneous states with local translation symmetry, described by
the order parameter (OP) with local transformation properties k=k(X). It is
shown that in the OP extended derivative we should consider compensating field
which is equivalent to the distortion tensor. The definition of the phonon
potential as a 4-distortion tensor is given and equations of motion of a
particle in a phonon field are obtained. We construct the Ginzburg-Landau
potential, which clearly describes the electron-phonon interaction and gives
the correct definition of the quantum of magnetic flux without doubling the
phenomenological charge interaction. The description of the states with the
d-symmetry of the wave superconducting gap by the lowsymmetric solutions is
suggested, in which the inversion of space-time is not equivalent to the OP
complex conjugation. A comparative analysis of the nonlinear Landau theory with
the compensating field distortion and the Kadi\'c-Edelen linear gauge theory of
dislocations is performed. It was demonstrated that the minimal interaction is
a consequence of local wave properties of OP, which is consistent with the
basic principles of quantum mechanics.
