  We describe some one-dimensional moduli spaces of rank 2 Gieseker semistable
sheaves on an Enriques surface improving earlier results of H. Kim. In case of
a nodal Enriques surface the obtained moduli spaces are reducible for general
polarizations. For unnodal Enriques surfaces we show how to reduce the study of
moduli spaces of high even rank Gieseker semistable sheaves to low ranks. To
prove this we use the method of K. Yoshioka who showed that in the odd rank one
can reduce to rank 1.
