  In this paper we study affine hypersurfaces with non-degenerate second
fundamental form of arbitrary signature additionally equipped with an almost
symplectic structure $\omega$.
  We prove that if $R^p\omega=0$ or $\nabla^p\omega =0$ for some positive
integer $p$ then the rank of the shape operator is at most one. The results
provide complete classification of affine hypersurfaces with higher order
parallel almost symplectic forms and are generalization of recently obtained
results for Lorentzian affine hypersurfaces.
