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