  We discuss common properties and reasons for integrability in the class of
multistate Landau-Zener (MLZ) models with all diabatic levels crossing at one
point. Exploring the Stokes phenomenon, we show that each previously solved
model has a dual one, whose scattering matrix can be also obtained
analytically. For applications, we demonstrate how our results can be used to
study conversion of molecular into atomic Bose condensates during passage
through the Feshbach resonance, and provide purely algebraic solutions of the
bowtie and special cases of the driven Tavis-Cummings model (DTCM).
