through the region where $\dot \th \apx 0$, that we call a [*resonance zone*]{}. In the next two sections we develop a general theory of resonance crossings in volume-preserving systems analogous to the one first developed in [@Neishtadt:1997] and then adjusted in [@Neishtadt:1999] for Hamiltonian systems. The equation $\om(I_1, I_2) = 0$ defines a 2-D surface in the original 3-D space (a curve on the $(I_1, I_2)$ plane), that we call the [*resonant surface*]{}, or the [*resonance*]{}, and denote by ${\mit R}$. In the vicinity of a resonance the variable $\th$ is not fast compared with $I_1$ and $I_2$. Hence, we cannot expect averaged system (\[vp2\]) to approximate the exact system adequately. As a result, the value of the integral of the averaged system, $\Ffi$, may change in the process of a passing through the vicinity of the resonance.

Qualitatively a single passage through the vicinity of a resonance can be described as follows. A characteristic streamline of the exact system approaches the resonant zone with the value of $\Ffi$ oscillating with a small amplitude, $\sim \eps$, near some value $\Ffi^-$. When in the process of the motion it arrives to the resonant zone, it is either captured into the 