On the stability of linear systems with a quadratic integral

The problem of stability of non-degenerate linear systems admitting a first integral in the form of a non-degenerate quadratic form is considered. New algebraic criteria for stability, as well as complete instability of such systems, have been established in the form of equality to zero of traces of products of matrices, which include an additional symmetric matrix. These conditions are closely related to the symplectic geometry of the phase space, which is determined by the matrix of the original linear system and the symmetric matrix defining the first integral. General results are applied to finding conditions for complete instability of linear gyroscopic systems.