Exact S-matrices for higher dimensional representations of generalized Landau-Zener Hamiltonians

We explore integrable Landau-Zener-type Hamiltonians through the framework of Lie algebraic structures. By reformulating the classic two-level Landau-Zener model as a Lax equation, we show that higher-spin generalizations lead to exactly solvable scattering matrices, which can be computed efficiently for any higher-spin representation. We further extend this approach to generalized bow-tie Landau-Zener Hamiltonians, employing non-Abelian gauge fields that satisfy a zero-curvature condition to derive their scattering matrices algebraically. This method enables the systematic construction of new exactly solvable multi-level models; as a result, we present previously unknown six-dimensional and eight-dimensional Landau-Zener models.