On quadratic integral Poincare–Zhukovsky's equations

For Poincaré–Zhukovsky's equations with non-diagonal matrices in the Hamiltonian, we obtain conditions for existence of the quadratic integral $(\mathbf{YS}, \mathbf{K}) = \mathrm{const}$ and the explisit form of it. It is shown that if the integral exists, then the equations reduce to the Schottky's case.