When the $n$-sphere, $S^n$,admit a real polarization $D\subset TS^n$



$S^n$ admit a real polarization $D\subset TS^n$?
Well, this only would make sense when $n$ is even, but there are two problems: First, except when $n=1$, the $2n$-sphere does not carry any symplectic structure. Second, the tangent bundle of the $2n$-sphere has no nontrivial subbundles anyway. (The reason is that $TS^{2n}$ has nonzero Euler class, so it cannot be written as $P\oplus Q$ where $P$ and $Q$ have positive rank, since these two bundles would have to be orientable and have zero Euler class.)


