  We provide an existence theory for gradient Gibbs measures for Z-valued spin
models on regular trees which are not invariant under translations of the tree,
assuming only summability of the transfer operator. The gradient states we
obtain are delocalized. The construction we provide for them starts from a
two-layer hidden Markov model representation in a setup which is not invariant
under tree-automorphisms, involving internal q-spin models. The proofs of
existence and lack of translation invariance of infinite-volume gradient states
are based on properties of the local pseudo-unstable manifold of the
corresponding discrete dynamical systems of these internal models, around the
free state, at large q.
