A two-dimensional problem of phase transitions in continuum mechanics with identical elastic modules

We give a sufficient condition for the unsolvability of the two-dimensional problem of phase transitions in the mechanics of two-phase elastic media with identical tensors of elastic moduli. It is of geometric nature and is based on the properties of the set of points of contact of the ellipsoid in the three-dimensional space of $2\times2$-symmetric matrices with the conical surface in this space. The ellipsoid depends on the elasticity moduli tensor and the difference of residual deformation tensors while the conical surface consists of degenerate matrices.