  Let $\A$ be an arrangement of affine lines in $\C^2,$ with complement
$\M(\A).$ The (co)homo-logy of $\M(\A)$ with twisted coefficients is strictly
related to the cohomology of the Milnor fibre associated to the conified
arrangement, endowed with the geometric monodromy. Although several partial
results are known, even the first Betti number of the Milnor fiber is not
understood. We give here a vanishing conjecture for the first homology, which
is of a different nature with respect to the known results. Let $\Gamma$ be the
graph of \emph{double points} of $\A:$ we conjecture that if $\Gamma$ is
connected then the geometric monodromy acts trivially on the first homology of
the Milnor fiber (so the first Betti number is combinatorially determined in
this case). This conjecture depends only on the combinatorics of $\A.$ We prove
it in some cases with stronger hypotheses.
  In the final parts, we introduce a new description in terms of the group
given by the quotient ot the commutator subgroup of $\pi_1(\M(\A))$ by the
commutator of its \emph{length zero subgroup.} We use that to deduce some new
interesting cases of a-monodromicity, including a proof of the conjecture under
some extra conditions.
