  We explicitly construct Brill--Noether general $K3$ surfaces of genus $4,6$
and $8$ having the maximal number of elliptic pencils of degrees $3, 4$ and
$5$, respectively, and study their moduli spaces and moduli maps to the moduli
space of curves. As an application we prove the existence of Brill--Noether
general $K3$ surfaces of genus $4$ and $6$ without stable Lazarsfeld--Mukai
bundles of minimal $c_2$.
