Criterion on the nonexistence of exact invariants in adiabatic systems

In the present paper we investigate the nonintegrability of adiabatic one degree of freedom Hamiltonian systems, with the additional assumption that the frozen system possesses an unstable fixed point with two asymmetric homoclinic loops. We prove a criterion for the nonexistence of an integral for such systems, and therefore we prove the nonexistence of a quantity which is conserved in an arbitrarily high order on epsilon. A specific application is given in the asymmetric quartic oscillator with adiabatic time dependence.