Heavy particle collisions, in particular low-energyion-atom collisions, are amenable to semiclassical JWKBphase integral analysis in the complex plane of theinternuclear separation. Analytic continuation in thisplane requires due attention to the Stokes phenomenonwhich parametrizes the physical mechanisms of curvecrossing, non-crossing, the hybrid Nikitin model,rotational coupling and predissociation. Complextransition points represent adiabatic degeneracies. Inthe case of two or more such points, the Stokes constantsmay only be completely determined by resort to theso-called comparison-equation method involving, inparticular, parabolic cylinder functions or Whittakerfunctions and their strong-coupling asymptotics. Inparticular, the Nikitin model is a two transition-pointone-double-pole problem in each half-plane correspondingto either ingoing or outgoing waves. When the fourtransition points are closely clustered, new techniquesare required to determine Stokes constants. However, suchinvestigations remain incomplete. A model problem istherefore solved exactly for scattering along aone-dimensional z-axis. The energy eigenvalue is b2-a2 and the potential comprises -z2/2(parabolic) and -a2 + b2/2z2(centrifugal/centripetal) components. The square of thewavenumber has in the complex z-plane, four zeros eacha transition point at z = ±a ±ib and has adouble pole at z = 0. In cases (a) and (b), a andb are real and unitarity obtains. In case (a) the reflection and transitioncoefficients are parametrized by exponentials when a2 + b2 >½. In case (b) they are parametrized by trigonometrics when a2 + b2 <½ and total reflection is achievable. In case (c) a andb are complex and in general unitarity is not achieved due to loss of fluxto a continuum (O'Rourke and Crothers, 1992 Proc. R. Soc. 438 1). Nevertheless, case (c)coefficients reduce to (a) or (b) under appropriatelimiting conditions. Setting z = λτ, withλ a real constant, an attempt is made to model a two-statecollision problem modelled by a pair of coupled first-order impactparameter equations and an appropriateT̃-τ relation, where T̃ is theStueckelberg variable and τ is the reduced or scaledtime. The attempt fails because T̃ is an oddfunction ofτ, which is unphysical in a real collision problem. However, it ispointed out that by applying the Kummer exponential model to eachhalf-plane (O'Rourke and Crothers 1994 J. Phys. B:At. Mol. Opt. Phys. 27 2497) the current modelis in effect extended to a collision problem with fourtransition points and a double pole in each half-plane.Moreover, the attempt in itself is not a complete failuresince it is shown that the result is a perfect diabaticinelastic collision for a traceless Hamiltonian matrix,or at least when both diagonal elements are odd and theoff-diagonal elements equal and even.