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We construct colored lattice models whose partition functions representsymplectic and odd orthogonal Demazure characters and atoms. We show that ourlattice models are not solvable, but we are able to show the existence ofsufficiently many solutions of the Yang--Baxter equation that allows us tocompute functional equations for the corresponding partition functions. Fromthese functional equations, we determine that the partition function of ourmodels are the Demazure atoms and characters for the symplectic and oddorthogonal Lie groups. We coin our lattice models as quasi-solvable. We use thenatural bijection of admissible states in our models with Proctor patterns togive a right key algorithm for reverse King tableaux and Sundaram tableaux.

Quasi-solvable lattice models for $\mathrm{Sp}_{2n}$ and $\mathrm{SO}_{2n+1}$ Demazure atoms and characters