  We analyse an exactly solvable spin-$1/2$ chain which is a generalised
version of Kitaev's honeycomb model. We show that every state of the system has
a $2^{N/4}$ fold degeneracy, where $N$ is the number of sites. We present
analytic solutions for the zero energy modes of the Majorana fermions.
Localised, unpaired Majorana modes occur even in the bulk of the chain and they
are bound to kink (anti-kink) $Z_2$ flux configurations. The unpaired Majorana
modes can therefore be created and manipiulated if the $Z_2$ flux
configurations can be controlled. We delineate the regions in parameter space
for homogenous chains where the zero modes occur. We further show that there is
a large parameter space for inhomogenous chains where the unpaired modes occur
and that their wavefunctions can be tuned if the couplings of the model can be
tuned.
