Let $M$ be a real, even dimensional, compact manifold endowed with a symplectic form $\omega$ and a flat, torsionless connection $\nabla$ compatible with $\omega$, that is $$\nabla \omega=0.$$

Under this assumptions, taking into account a simplified version of the Markus conjecture, it should follow that $M$ is geodesically complete.

*My question is: Under what additional geometric assumptions is the result known to be true?*

One concrete example is: If in addition we assume the existence of a parallel almost complex structure $J$ on $M,$ then the result is true for $2$ dimensional and $4$ dimensional manifolds as observed by Misha.

What if instead of the the parallel almost complex structure we assume that there is a nonzero parallel vector field $V$ on $M?$ Does it follow that the manifold is geodesically complete?