Let $X\rightarrow S$ be a (projective, flat,... or any other assumption which makes you happy) fibration of a smooth threefold over a smooth surface with connected one-dimensional fibers.

As an example you may consider a generic hypersurface of bi-degree (3,3) in $\mathbb{P}^2\times \mathbb{P}^2$ with the fibration induced from either of fibrations $\mathbb{P}^2\times \mathbb{P}^2 \rightarrow \mathbb{P}^2 $.

My question is, how we can find whether this fibrations has a section or not? Are there any known methods which might help us to answer this question for a given fibration? Or any obstruction?

Remark: It is a known theorem that any elliptic fibration with section is a resolution of a Weierstrass model, but for a given elliptic fibration as above, I don't know how to find whether it has a section or not.