  Let $(C,\underline{w})$ be a polarized nodal reducible curve. In this paper
we consider coherent systems of type $(r,d,k)$ on $C$ with $k < r$. We prove
that the moduli spaces of $(\underline{w},\alpha)$-stable coherent systems
stabilize for large $\alpha$ and we generalize several results known for the
irreducible case when we chose a good polarization. Then, we study in details
the components of moduli spaces containing coherent systems arising from
locally free sheaves.
