  We consider the Ising model on a Cayley tree of order two with nearest
neighbor interactions and competing next nearest neighbor interactions
restricted to spins belonging to the same branch of the tree. This model was
studied by Vannimenus and found a new modulated phase, in addition to the
paramagnetic, ferromagnetic, antiferromagnetic phases and a (+ + - -) periodic
phase. Vannimenus's results based on the recurrence equations (relating the
partition function of an $n-$ generation tree to the partition function of its
subsystems containing $(n-1)$ generations) and most results are obtained
numerically. In this paper we analytically study the recurrence equations and
obtain some exact results: critical temperatures and curves, number of several
phases, partition function.
