= Solution
\b[False.] Give $E=TS^2\to S^2$ a fibrewise area form. This is a rank-two <symplectic vector bundle>. A <Lagrangian subbundle> would be a real line subbundle $L\subset TS^2$. Real <line bundles> are classified by their first <Stiefel–Whitney class> in $H^1(B;\mathbb Z/2)$; since this group vanishes for $S^2$, $L$ would be trivial. A nonzero section of $L$ would therefore be a nowhere-vanishing tangent <vector field> on $S^2$. The <Hairy ball theorem> rules this out: every continuous tangent <vector field> on an even-dimensional sphere has a zero. Hence $TS^2$ does not even have one <Lagrangian subbundle>, let alone a splitting into two. Fibrewise existence of <Lagrangian subspaces> does not guarantee a globally continuous choice.
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