  In this paper, we consider the Ising-Vanniminus model on an arbitrary order
Cayley tree. We generalize the results conjectured in [Chinese Journal of
Physics, 54 (4), 635-649 (2016)] and [International Journal of Modern Physics,
arXiv:1608.06178] for an arbitrary order Cayley tree. We establish existence
and a full classification of translation invariant Gibbs measures with memory
of length 2 associated with the model on arbitrary order Cayley tree. We
construct the recurrence equations corresponding generalized ANNNI model. We
satisfy the Kolmogorov \emph{consistency} condition. We propose a rigorous
measure-theoretical approach to investigate the Gibbs measures with memory of
length 2 for the model. We explain whether the number of branches of tree does
not change the number of Gibbs measures. Also we take up with trying to
determine when phase transition does occur.
