  We construct two types of non-holomorphic Lefschetz fibrations over $S^2$
with $(-1)$-sections ---hence, they are fiber sum indecomposable--- by giving
the corresponding positive relators. One type of the two does not satisfy the
slope inequality (a necessary condition for a fibration to be holomorphic) and
has a simply-connected total space, and the other has a total space that cannot
admit any complex structure in the first place. These give an alternative
existence proof for non-holomorphic Lefschetz pencils without Donaldson's
theorem.
