  This paper is the second in a series. The first one describes pillow
degenerations of a $K3$ surface with genus $g$. In this paper we study the
$(2,2)$-pillow degeneration of a non-prime $K3$ surface and the braid monodromy
of the branch curve of the surface with respect to a generic projection onto
$\C\P^2$. In future papers we study the fundamental group of the complement of
the branch curve and the fundamental group of the Galois cover of the surface
with respect to this generic projection.
