  Ising model with competing nearest-neighbors and prolonged
next-nearest-neighbors interactions on a Cayley tree has long been studied but
there are still many problems untouched. This paper tackles new Gibbs measures
of Ising-Vannimenus model with competing nearest-neighbors and prolonged
next-nearest-neighbors interactions on a Cayley tree (or Bethe lattice) of
order three. By using a new approach, we describe the translation-invariant
Gibbs measures for the model. We show that some of the measures are extreme
Gibbs distributions. In this paper we take up with trying to determine when
phase transition does occur.
