  We prove that any genus-2 Lefschetz fibration without reducible fibers and
with ``transitive monodromy'' is holomorphic. The latter condition comprises
all cases where the number of singular fibers is not congruent to 0 modulo 40.
An auxiliary statement of independent interest is the holomorphicity of
symplectic surfaces in S^2-bundles over S^2, of relative degree up to 7 over
the base, and of symplectic surfaces in CP^2 of degree up to 17.
