Splitting of the separatrices and the nonexistence of first integrals in systems of differential equations of Hamiltonian type with two degrees of freedom

An investigation is made of the phenomenon of splitting of the real and complex separatrices of hyperbolic cycles of systems of differential equations of Hamiltonian type with two degrees of freedom, and of its connection with the absence of additional meromorphic first integrals for these systems. The results obtained are used to prove the absence of a nonconstant meromorphic first integral in a system describing a stationary flow of an ideal incompressible fluid with periodic boundary conditions and with velocity field collinear with its curl.
Bibliography: 15 titles.