Аннотация:An ODE system of order 5 called the two-system of the one-dimensional anharmonic oscillator is considered. The system is Hamiltonian with respect to a degenerate Poisson bracket and can be viewed as a 1-parameter family of canonical systems with 2 d.o.f. on symplectic leaves, and also as a perturbation of an integrable system, degenerate in the sense of the KAM theory. Neither the KAM theory, nor the Birkhoff normal form are directly applicable, and the perturbation analysis is carried out by means of a non-Hamiltonian normal form. The system possesses a formal integral in any order of the perturbation theory and thus long-persistent adiabatic tori. The analytical description of such tori, a place of exact periodic orbits between them, and supporting numerical results are presented.