  We pose the problem of metastability for a three--state spin system with
conservative dynamics. We consider the Blume--Capel model with the Kawasaki
dynamics, we prove that, in a particular region of the parameter plane, the
metastable state is the unique homogeneous minus state, and we estimate the
exit time. To achieve our goal we have to solve several variational problems in
the configuration space which result to be particularly involved, due to
complicated structure of the trajectories. They key ingredient is the control
of the energy differences between the configurations crossed when a spin is
transported from the boundary to an internal site of the lattice through a
completely arbitrary mixture of the three--state spin species. To master these
mechanisms we have introduced a new approach based on the transport of spins
along nearest neighbor connected regions of the lattice with constant spin
configuration. This novel approach goes beyond the Blume--Capel model and can
be used for the study of more general multi--state spin models.
