  We study a one-dimensional version of the Kitaev model on a ring of size N,
in which there is a spin S > 1/2 on each site and the Hamiltonian is J \sum_i
S^x_i S^y_{i+1}. The cases where S is integer and half-odd-integer are
qualitatively different. We show that there is a Z_2 valued conserved quantity
W_n for each bond (n,n+1) of the system. For integer S, the Hilbert space can
be decomposed into 2^N sectors, of unequal sizes. The number of states in most
of the sectors grows as d^N, where d depends on the sector. The largest sector
contains the ground state, and for this sector, for S=1, d =(\sqrt{5}+1)/2. We
carry out exact diagonalization for small systems. The extrapolation of our
results to large N indicates that the energy gap remains finite in this limit.
In the ground state sector, the system can be mapped to a spin-1/2 model. We
develop variational wave functions to study the lowest energy states in the
ground state and other sectors. The first excited state of the system is the
lowest energy state of a different sector and we estimate its excitation
energy. We consider a more general Hamiltonian, adding a term \lambda \sum_n
W_n, and show that this has gapless excitations in the range \lambda^c_1 \leq
\lambda \leq \lambda^c_2. We use the variational wave functions to study how
the ground state energy and the defect density vary near the two critical
points \lambda^c_1 and \lambda^c_2.
