of the paper (Sects. 2–4) we present a general theory of passages through resonances in near-integrable volume-preserving 3-D systems. In other words, we study the flows that are a superposition of an integrable base flow and a weak perturbation. While the inclusion of the time-dependence adds to the variety of possible flow configurations (see e.g. [@CartwrightFeingoldandPiro:1995; @Solomon:2003]) and extends the range of admissible resonances, we will see that our (time-independent) setting is sufficient to observe most of the related phenomena and is better suited for analytical description.

In the present paper we consider the base flows belonging to a class of integrable 3-D flows that possesses two invariants. Introducing these invariants as new variables we can write the evolution equations in the following form: \_1 &=& 0,\
\_2 &=& 0,\
&=& ( I\_1, I\_2 ). In such flows almost all the streamlines are closed curves (except for those that connect the stationary points). Each curve is defined by the values of $I_1$ and $I_2$ and we denote it as $\Gm_{I_1, I_2}$. Such systems belong to the two actions and one angle class of dynamical systems, where a single changing variable being a coordinate along a streamline.

A generic small perturbation 