The spectra of linear Hamiltonian system and the symplectic geometry of complex Artin space

The spectrum of the linear Hamiltonian system is considered in the complex phase space C2n, which contains the smooth submanifold of singular planes S and the smooth submanifold of Lagrangian planes. It turns out that the number of points of S and L plane intersection and the structure of singular Lagrangian planes are closely connected with the structure of the linear Hamiltonian system spectrum. It is proved that if all eigenvalues of the Hamiltonian system are prime then S and L submanifolds intersect equally in 2n different points.