  We consider the SOS (solid-on-solid) model, with spin values $0,1,2$, on the
Cayley tree of order two (binary tree). We treat both ferromagnetic and
antiferromagnetic coupling, with interactions which are proportional to the
absolute value of the spin differences.
  We present a classification of all translation-invariant phases (splitting
Gibbs measures) of the model: We show uniqueness in the case of
antiferromagnetic interactions, and existence of up to seven phases in the case
of ferromagnetic interactions, where the number of phases depends on the
interaction strength.
  Next we investigate whether these states are extremal or non-extremal in the
set of all Gibbs measures, when the coupling strength is varied, whenever they
exist.
  We show that two states are always extremal, two states are always
non-extremal, while three of the seven states make transitions between
extremality and non-extremality. We provide explicit bounds on those transition
values, making use of algebraic properties of the models, and an adaptation of
the method of Martinelli, Sinclair, Weitz.
