On the spectral properties of the pencil of Monge-Amper non-linear equations in vector-functions spaces

The eigenvalue problem on Monge-Amper non-linear system of differential equations in Hilbert space of vector-functions has been considered in the article. The connection of this problem with the well-known Sobolev-Alexandrian operator has been revealed and the finite multiplicity and the real ness of the eigenvalues have been proved. The eigenvalues and the system of eigen vector-functions are given in explicit form, when the domain is a unit circle.