The linear curve-crossing problem involving a slanted potential and a set of horizontal ones as an exactly soluble quantum-mechanical problem, is re-examined. Special attention is paid to the questions of degeneracy of the horizontal-potential channels. It is shown, by explicit derivation of an asymptotic expansion of the scattering amplitudes, that the degeneracy limit is a singular point. An explicit exact solution is derived for the degenerate case, resulting in a non-Landau-Zener saturated behaviour of the transition probabilities at high coupling intensities. Estimates are derived for the transition region as a function of the energy gap, and its divergence on approach to degeneracy. Relevance to the optical shielding of ultracold atom collisions is pointed out, with particular reference to the phenomenon of 'counterintuitive' transitions which were discussed recently.

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