Context. When a subhalo interacts with a cold stellar stream, the otherwise nearly smooth distribution of stars is disturbed, and this creates a gap. The properties of these gaps depend on the interaction parameters. Their characterisation could thus lead to a determination of the mass spectrum of the perturbers and might reveal the existence of dark subhalos orbiting the Milky Way. Aims. Our goal is to construct a fully analytical model of the formation and evolution of gaps embedded in streams orbiting in a realistic Milky Way potential. Methods. To this end, we extended our previous model for spherical potentials and predict the properties of gaps in streams evolving in axisymmetric Stäckel potentials. We used action-angles and their simple behaviour to calculate the divergence of initially nearby orbits that are slightly perturbed by the interaction with a subhalo. Results. Our model, corroborated by N-body experiments, predicts that the size of a gap grows linearly with time. We obtain analytical expressions for the dependences of the growth rate on the orbit of the stream, the properties of the subhalo (mass and scale radius), and the geometry of the encounter (relative velocity and impact parameter). We find that the density at the centre of the gap decreases with time as a power law in the same way as the density of a stream. This causes the density contrast between a pristine and a perturbed stream on the same orbit to asymptotically reach a constant value that only depends on the encounter parameters. Conclusions. We find that at a fixed age, smallish gaps are sensitive mostly to the mass of the subhalo, while gaps formed by subhalo flybys with a low relative velocity, or when the stream and subhalo move in parallel, are degenerate to the encounter parameters.