Eigenvalue problem for some tensors used in mechanics and a number of essential compatibility conditions for the Saint-Venant deformation

Several questions related to the problem on the eigenvalues of the tensor $\begin{smallmatrix} \displaystyle{}\\ \overset{\vphantom{p}}{\stackrel{\displaystyle\mathbf A}{\stackrel{\sim}{\sim}}}\\ \end{smallmatrix}\in\mathbb R_4(\Omega)$ with special symmetries are considered. Here $\Omega$ is a certain region of, in general, four-dimensional (three-dimensional) Riemann space. It is proved that in this case a non-degenerate tensor of the fourth rank in the case of a four-dimensional (three-dimensional) Riemann space has no more than six (three) essential components. It is shown that the number of essential conditions of deformation Saint-Venant compatibility less than six.