  A rank one local system $\LL$ on a smooth complex algebraic variety $M$ is
admissible roughly speaking if the dimension of the cohomology groups
$H^m(M,\LL)$ can be computed directly from the cohomology algebra $H^*(M,\C)$.
  We say that a line arrangement $\A$ is of type $\CC_k$ if $k \ge 0 $ is the
minimal number of lines in $\A$ containing all the points of multiplicity at
least 3. We show that if $\A$ is a line arrangement in the classes $\CC_k$ for
$k\leq 2$, then any rank one local system $\LL$ on the line arrangement
complement $M$ is admissible. Partial results are obtained for the class
$\CC_3$.
