  Let M be a compact, connected non-orientable surface without boundary and of
genus g greater than or equal to 3. We investigate the pure braid groups P_n(M)
of M, and in particular the possible splitting of the Fadell-Neuwirth short
exact sequence 1 --> P_m(M {x_1,...,x_n}) --> P_{n+m}(M) --> P_n(M) --> 1,
where m,n are positive integers, and the homomorphism p*:P_{n+m}(M) --> P_n(M)
corresponds geometrically to forgetting the last m strings. This problem is
equivalent to that of the existence of a section for the associated fibration
p:F_{n+m}(M)} --> F_n(M) of configuration spaces, defined by
p((x_1,...,x_n,..., x_{n+m}))= (x_1, ..., x_n). We show that p and p* admit a
section if and only if n=1. Together with previous results, this completes the
resolution of the splitting problem for surfaces pure braid groups.
