  We study the two main types of trajectories of the ABC flow in the
near-integrable regime: spiral orbits and edge orbits. The former are helical
orbits which are perturbations of similar orbits that exist in the integrable
regime, while the latter exist only in the non-integrable regime. We prove
existence of ballistic (i.e., linearly growing) spiral orbits by using the
contraction mapping principle in the Hamiltonian formulation, and we also find
and analyze ballistic edge orbits. We discuss the relationship of existence of
these orbits with questions concerning front propagation in the presence of
flows, in particular, the question of linear (i.e., maximal possible) front
speed enhancement rate for ABC flows.
