  In this paper we consider translation-invariant Gibbs measures for the
Blum-Kapel model on a Cayley tree of order k. An approximate critical
temperature T_{cr} is found such that for T\geq T_{cr} there exists a unique
translation-invariant Gibbs measure and for 0<T<T_{cr} there are exactly three
translation-invariant Gibbs measures. In addition, we studied the problem of
(not) extremality for the unique Gibbs measure.
