  We study the rigidity properties of a class of algebraic Z^3-actions with
entropy rank two. For this class, conditions are found which force an invariant
measure to be the Haar measure on an affine subset. This is applied to show
isomorphism rigidity for such actions, and to provide examples of
non-isomorphic Z^3-actions with all their Z^2-sub-actions isomorphic. The
proofs use lexicographic half-space entropies and total ergodicity along
critical directions.
